Jesus rejects equality with God - How does this statement reconcile with traditional teachings?

It means that Jesus is God incarnate. God, or the Father, is simply pure consciousness and purely every unmanifested idea or thing. Therefore Jesus is the second to God, or the physical manifestation of the Father, which is pure potential. God is made manifest by and through man, or Jesus. Creation is finished, and God is made manifest through the thoughts of man. This manifestation is named Jesus/Yeheshua etc.

Athanasian Creed

According to the Athanasian Creed:

Whosoever will be saved, before all things it is necessary that he hold the catholic faith. Which faith except every one do keep whole and undefiled; without doubt he shall perish everlastingly. And the catholic faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence. For there is one Person of the Father; another of the Son; and another of the Holy Ghost. But the Godhead of the Father, of the Son, and of the Holy Ghost, is all one; the Glory equal, the Majesty coeternal. Such as the Father is; such is the Son; and such is the Holy Ghost. The Father uncreated; the Son uncreated; and the Holy Ghost uncreated. The Father unlimited; the Son unlimited; and the Holy Ghost unlimited. The Father eternal; the Son eternal; and the Holy Ghost eternal. And yet they are not three eternals; but one eternal. As also there are not three uncreated; nor three infinites, but one uncreated; and one infinite. So likewise the Father is Almighty; the Son Almighty; and the Holy Ghost Almighty. And yet they are not three Almighties; but one Almighty. So the Father is God; the Son is God; and the Holy Ghost is God. And yet they are not three Gods; but one God. So likewise the Father is Lord; the Son Lord; and the Holy Ghost Lord. And yet not three Lords; but one Lord. For like as we are compelled by the Christian verity; to acknowledge every Person by himself to be God and Lord; So are we forbidden by the catholic religion; to say, There are three Gods, or three Lords. The Father is made of none; neither created, nor begotten. The Son is of the Father alone; not made, nor created; but begotten. The Holy Ghost is of the Father and of the Son; neither made, nor created, nor begotten; but proceeding. So there is one Father, not three Fathers; one Son, not three Sons; one Holy Ghost, not three Holy Ghosts. And in this Trinity none is before, or after another; none is greater, or less than another. But the whole three Persons are coeternal, and coequal. So that in all things, as aforesaid; the Unity in Trinity, and the Trinity in Unity, is to be worshipped. He therefore that will be saved, let him thus think of the Trinity.

Furthermore, it is necessary to everlasting salvation; that he also believe faithfully the Incarnation of our Lord Jesus Christ. For the right Faith is, that we believe and confess; that our Lord Jesus Christ, the Son of God, is God and Man; God, of the Substance [Essence] of the Father; begotten before the worlds; and Man, of the Substance [Essence] of his Mother, born in the world. Perfect God; and perfect Man, of a reasonable soul and human flesh subsisting. Equal to the Father, as touching his Godhead; and inferior to the Father as touching his Manhood. Who although he is God and Man; yet he is not two, but one Christ. One; not by conversion of the Godhead into flesh; but by assumption of the Manhood into God. One altogether; not by confusion of Substance [Essence]; but by unity of Person. For as the reasonable soul and flesh is one man; so God and Man is one Christ; Who suffered for our salvation; descended into hell; rose again the third day from the dead. He ascended into heaven, he sitteth on the right hand of the God the Father Almighty, from whence he will come to judge the living and the dead. At whose coming all men will rise again with their bodies; And shall give account for their own works. And they that have done good shall go into life everlasting; and they that have done evil, into everlasting fire. This is the catholic faith; which except a man believe truly and firmly, he cannot be saved.

So the creed says that the Son and the Father are co-equal, and that neither are greater than or lesser than the other. According to the trinity doctrine, Jesus and the Father are h*m*oúsios, or 'of the same substance'. This was a word used by a few gnostic sects prior to Nicea, but it's basically a made-up word to describe the 'relationship' of the 'three divine persons'.

Nicene Creed

In 341 AD, at the Council of Nicea, it was decided that Jesus is 'God the Son', the 'second person' of the 'triune God'; equal and 'eternally begotten' of the Father. However, there were still many people that found this hard to accept. One of the main problems was, Yeshua says:

"the father is greater than I"

...as well as many, many, many other things that Yeshua and the Apostles said.

Council of Ephesus

In 431 AD, at the Council of Ephesus, the church made up a different term: Hypostatic Union. During this council, Cycil of Alexandria said:

“We must follow these words and teachings, keeping in mind what ‘having been made flesh’ means...that the Word, by having united to himself hypostatically flesh animated by a rational soul, inexplicably and incomprehensibly became man.” Letters 1-50 (The Fathers of the Church, Vol. 76; pg. 39)

So there you have it. It is an incomprehensible mystery why Yeshua talked as though the father was greater than himself. Sometimes Jesus talked like a man using his 'human nature', and sometimes he spoke as God, using his 'divine nature". If anything Yeshua says seems to contradict that he is God, just put that under the 'human nature' category and you can ignore it.

Conclusion

According to Christianity, the one true God exists as an incomprehensible mystery. He has commanded his believers to explain this incomprehensible mystery to others, and he has determined that every human that has ever existed must believe in this incomprehensible God. If somebody fails to comprehend this, just make up words and tell them that the made-up words explain everything. If any person that has ever existed fails to accept this incomprehensible mystery, and the many terms associated with it that are found nowhere in any bible ever written- God will cast them into an eternal hell to have their flesh burned forever and ever in agonizing pain.

There is no conflict.

Satan, and under temptation, Adam and Eve, grasped at equality with God. That was pride. It was usurpation of what was not theirs.

Jesus did not grasp at equality. This is not an admission of inequality, but a posture of humility (Matthew 11:29). Though being equal with God (John 10:30), he chose not to lay it on thick, demand his rights, call on angels to rescue him (Matthew 4:6, 26:53), claim his kingdom immediately (Matthew 4:8), and Lord it over others (Matthew 20:25). He chose to be a servant (Matthew 12:18, Philippians 2:7), though entitled to insist he be treated as king (Isaiah 9:6, Revelation 1:5).

In Ecclesiastes, Solomon addresses a conundrum. Some people inherit their kingdom, while others rise from lowly beginnings, even prison to attain kingship (Ecclesiastes 4:14). Sometimes the inheritor of a kingdom is a good and noble ruler, but sometimes is foolish, like Solomon's son Rehoboam (Ecclesiastes 2:18-19). So too the servant that rises to the kingship. It may go well for awhile, but people may eventually become disenchanted.

Jesus is both types of King. He inherited his kingdom, but also set aside his title, became a lowly servant, entered prison, and thereafter will come to claim his throne. This paradox of one who is both king by birth and also rises to the kingship from prison and obscurity is found throughout scripture. These are the two aspects of Jesus' nature, divine and human, coming together in one person. In order for the special qualitites of Christ to be visible to us, we needed to see him as primarily human in some situations and as primarily divine in others. God shines a flashlight so we can see the parts, but in eternity, we see the whole, with neither human nor divine overshadowed anymore.

Your question seems to be in some degree to be based on the hypostatic union.

Hypostatic union is a technical term in Christian theology employed in mainstream Christology to describe the union of Christ's humanity and divinity in one hypostasis, or individual existence. - Wikipedia.

Since Jesus is by nature both divine and human (both God and man) it should not be surprising that Our Lord spoke in ways that reflected one nature at one time and then in a way that reflected the another nature. When Jesus said that the "Father is greater than I", he was speaking through his human nature.

Jesus is the most important person who has ever lived since he is the Savior, God in human flesh. He is not half God and half man. He is fully divine and fully man. In other words, Jesus has two distinct natures: divine and human. Jesus is the Word who was God and was with God and was made flesh (John 1:1, 14). This means that in the single person of Jesus he has both a human and divine nature, God and man. The divine nature was not changed when the Word became flesh (John 1:1, 14). Instead, the Word was joined with humanity (Col. 2:9). Jesus' divine nature was not altered. Also, Jesus is not merely a man who "had God within Him," nor is he a man who "manifested the God principle." He is God in flesh, second person of the Trinity. "The Son is the radiance of God's glory and the exact representation of his being, sustaining all things by his powerful word." (Heb. 1:3). Jesus' two natures are not "mixed together" (Eutychianism), nor are they combined into a new God-man nature (Monophysitism). They are separate yet act as a unit in the one person of Jesus. This is called the Hypostatic Union.

The following chart should help you see the two natures of Jesus "in action":

GOD
He is worshiped (Matt. 2:2, 11; 14:33)

He was called God (John 20:28; Heb. 1:8)

He was called Son of God (Mark 1:1)

He is prayed to (Acts 7:59)

He is sinless (1 Pet. 2:22; Heb. 4:15)

He knows all things (John 21:17)

He gives eternal life (John 10:28)

All the fullness of deity dwells in Him (Col. 2:9)

MAN
He worshiped the Father (John 17)

He was called man (Mark 15:39; John 19:5)

He was called Son of Man (John 9:35-37)

He prayed to the Father (John 17)

He was tempted (Matt. 4:1)

He grew in wisdom (Luke 2:52)

He died (Rom. 5:8)

He has a body of flesh and bones (Luke 24:39)

The Communicatio Idiomatum

A doctrine that is related to the Hypostatic Union is the communicatio idiomatum (Latin for "communication of properties"). This is the teaching that the attributes of both the divine and human natures are ascribed to the one person of Jesus. This means that the man Jesus could lay claim to the glory He had with the Father before the world was made (John 17:5), claim that He descended from heaven (John 3:13), and also claim omnipresence (Matt. 28:20). All of these are divine qualities that are laid claim to by Jesus; therefore, the attributes of the divine properties were claimed by the person of Jesus.

For a proper understanding of Jesus and, therefore, all other doctrines that relate to Him, His two natures must be properly understood and defined. Jesus is one person with two natures. This is why He would grow in wisdom and stature (Luke 2:52) and yet know all things (John 21:17). He is the Divine Word that became flesh (John 1:1, 14). - Jesus' Two Natures: God and Man.

Jesus rejects equality with God - How does this statement reconcile with traditional teachings? It doesn't. The 'equality' teaching is indeed a tradition. If Christ was not convincing enough in John 17, other writings especially Isaiah 40:25 leave no room for options, as also Christ says that scripture cannot be broken.

Isaiah 40:25
"To whom will you compare me? Or who is my equal?" says the Holy One.

There are 2 things to qualify first before embarking on illustrating how Christ is, or not, 'equal' to the Father;

A. The term 'God'
B. The phrase 'equal with God'

A The term 'God'

In the spirit, essence is directly related to power.

Isaiah 45:5
I am the LORD, and there is none else, there is no God beside me: I girded thee, though thou hast not known me:

The Father is saying there are gods indeed, but none in all His creation is like Him.

Messiah says in John 10;

34 Jesus replied, “Is it not written in your Law: ‘I have said you are gods’?
35 If he(ultimately the Father) called them gods to whom the word of God came — and the Scripture cannot be broken
36 then what about the One whom the Father sanctified and sent into the world? How then can you accuse Me of blasphemy for stating that I am the Son of God? .

Now compare Psalm 8:5,

Yet You have made him a little lower than God, And You crown him with glory and majesty!

with;
Job 40:15, on the issue of 'making' man alongside that of another entity;

“Behold, Behemoth, which I made as I made you; he eats grass like an ox,

It's the same notion in Psalm 8:5 where man is contrasted to the entity 'god,' whose coming into existence is attributed to the Father just as man's is.

The inference is that there are 'gods' since the Father cannot contradict His word and all are because of Him, so often the term 'god' is an allusion to essense.

B Equal with God

The mist in the term equal with God then clears, making it to be true or false on the back of what scriptures above indicate as the position and nature of the Father, alongside that of gods who are by the same Father.
The question now is what do they mean by 'equality' if God in this phrase refers to the Father?

"Make them one even as we are One" and "I and the Father are one" don't make Messiah equal to the Father, rather, they depict both being of the same mind to achieve that for which Messiah is in flesh.
As also it should be obvious that the former saying doesn't mean that believers in Christ get to be equal.

Messiah is a highly distinguished 'Son of God', this makes Him God by essence, and when in the throne of the Father, He is God by title.

In Job 40, the Father describes the capabilities of an entity with a title of God.

9 Hast thou an arm like God? or canst thou thunder with a voice like him?
10 Deck thyself now with majesty and excellency; and array thyself with glory and beauty.
11 Cast abroad the rage of thy wrath: and behold every one that is proud, and abase him.
12 Look on every one that is proud, and bring him low; and tread down the wicked in their place.
13 Hide them in the dust together; and bind their faces in secret.
14 Then will I also confess unto thee that thine own right hand can save thee.

This is what is ascribed to entities labelled as Elohim, the same beings in Psalm 82:1

Yet even with all the preferment, Messiah categorically indicates many times that in His capacity He is only doing the Father's commandment and will. In the famous I am ascending to My Father and your Father, to My God and your God, as well as in Revelation, when Messiah and the Father are in the same circumstances as opposed to Him being in flesh while the Father was in heaven, Messiah still refers to the Father as 'my God'.

Revelation 3:2

Be alert and strengthen what remains, which is about to die, for I have not found your works complete before My God.

Messiah was given to believers to be 'worshipped. To 'worship' means essentially to fashion a heart's desires after a doctrine or way of life, and messiah's words aren't His words but the Father's ultimately. So, He's worshipped as the Father is also worshipped;

And in places like Hosea is where all this was foretold.

Hosea 3:5
5 Afterward the sons of Israel will return and seek the LORD their God and David their king; and they will come trembling to the LORD and to His goodness in the last days.

David is a messianic reference, as also the term seek is peculiar to issues of the heart of man. The same seeking is in;

Jeremiah 29:13
You will seek me and find me when you seek me with all your heart. SO the children of Israel in the days of this 'David' will seek David as well as the LORD at one and the same time in the last days, a time unique associated with the start of Messianic duties. In John 17 Messiah refers to the 'glory' that He had with the Father before the world begun, which existence before the world begun is mistaken for equality with the Father.

So Jesus rejects equality with God is a true statement as many scriptures indicate and therefore it conflicts with the traditional teaching of the 'Father being equal with His Servant.

Jesus is the Word of God made into flesh. The Word (Logos) is not primarily expression but the thought and intention behind the expression. As such Logos (the Word) exists prior to expression. In this way "the Word was with God" (John 1:1) prior to the Word being made flesh to dwell among us (John 1:14).

God has perfect integrity; i.e. he cannot or will not say or do anything that is in contradiction to who He is. God is also infinite therefore His integrity is infinite and there is no essential difference between who God is and what God says. As such Logos (the Word) "was God" (John 1:1).

God is eternal and cannot be imagined as ever being absent thought and intention. As such "In the beginning was the Word" (John 1:1)

This Word when expressed always accomplishes that purpose for which it was sent:

“For as the rain and the snow come down from heaven and do not return there but water the earth, making it bring forth and sprout, giving seed to the sower and bread to the eater, so shall my word be that goes out from my mouth; it shall not return to me empty, but it shall accomplish that which I purpose, and shall succeed in the thing for which I sent it. - Isaiah 55:10-11

And so the Word (Logos) when expressed can be understood as always in obedience to the God who expresses because it accomplishes God's purpose unfailingly and only.

For I have come down from heaven, not to do my own will but the will of him who sent me. And this is the will of him who sent me, that I should lose nothing of all that he has given me, but raise it up on the last day. For this is the will of my Father, that everyone who looks on the Son and believes in him should have eternal life, and I will raise him up on the last day.” - John 6:38-40 

As OP and many answers have pointed out, Jesus expresses His obedience, dependence upon, and subordination to the Father quite consistently. He is the Word of God expressed in flesh.

Wherever that expression is received it is received as though receiving God himself because the same perfect, infinite, eternal integrity shared by God's essence and His thoughts is also shared by His expression. Abraham, Samuel, Jeremiah, and so many more received the Word of the Lord just as if they received the Lord Himself.

Truly, truly, I say to you, whoever receives the one I send receives me, and whoever receives me receives the one who sent me.” - John 13:20

The Word of God before and after the expression in flesh (Jesus) is both equal to God in essence and subordinate to God in relationship. If the Word of God comes to us and we receive it as such, we receive God. Nothing has changed. Jesus is that Word of God who has come to us. If we receive Jesus as that Word, we receive God. If we receive Jesus as less than the Logos of God, we receive less than God:

“Whoever receives you receives me, and whoever receives me receives him who sent me. The one who receives a prophet because he is a prophet will receive a prophet's reward, and the one who receives a righteous person because he is a righteous person will receive a righteous person's reward. And whoever gives one of these little ones even a cup of cold water because he is a disciple, truly, I say to you, he will by no means lose his reward.” - Matthew 10:40-42

If we receive less than God then we have no life in us. 

I applaud Ken Graham's excellent answer about the dual nature of Jesus, but also allow me to present a slightly simplified answer from a different perspective.

The key verse in the ones quoted is

Philippians 2:6 (NAS) "although He existed in the form of God, did not regard equality with God a thing to be grasped"

In other words Jesus is God, but he voluntarily decided not to hold on the the status of being God. He relinquished the power, authority and status that he had by right (right of being God) and took up only such power, authority and status that God the Father chose to give him. He deliberately made himself less than God in status, while retaining his nature as being God.

You might find an analogy in the trope of a company owner who gets himself a job in the mailroom of his own company. If he does this properly, then he has the real status of a mailroom worker - he really does the work of a mailboy; the mailroom boss is really his boss, and he does what that mailboss tells him. On the other hand he is also fundamentally the owner of the company. Even if he choses not to exercise his rights as owner, the shares are in his name, and at some point in the future he will be able to take up the role of owner again. The man can rightfully say that he is the "servant" of the company owners, and thus "lesser" than them, even if one of the company owners is himself.

These beliefs have been common in Christianity for the best part of 2000 years, and have ample biblical support