How can Jesus be both root and offspring of David if he was only a man and did not pre-exist his incarnation?

Well, Though I believe Jesus preexisted his birth, I can provide away the verse be true without him doing so.

When considering Jesus as the root of David When you think "root of David" you may be thinking in reference to David's past human birth. However, Jesus has created for us a new birth in the resurrection. Indeed Jesus, himself born again, is declared begotten of God at his resurrection:

Acts 13:32-33 32 And we bring you the good news that what God promised to the fathers, 33 this he has fulfilled to us their children by RAISING Jesus, AS ALSO IT IS WRITTEN in the second Psalm,
“‘You are my Son, today I have begotten you.’

And indeed we do not fully become adopted into the family of God until our physical bodies are recreated:

Romans 8:23 23 And not only the creation but we ourselves, who have the firstfruits of the Spirit, groan inwardly as we wait eagerly for adoption as sons, the redemption of our bodies.

We are new creatures in Christ when we are reborn first spiritually but ultimately physically at the resurrection when God brings forth the New Heaven and New Earth.

2 Peter 3:13 13 But according to his promise we are waiting for new heavens and a new earth in which righteousness dwells.

Indeed we must be so recreated for our very physical existence is far too corrupt for the kingdom of God.

1 Corinthians 15:50-57 50 I tell you this, brothers: flesh and blood cannot inherit the kingdom of God, nor does the perishable inherit the imperishable. 51 Behold! I tell you a mystery. We shall not all sleep, but we shall all be changed, 52 in a moment, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, and the dead will be raised imperishable, and we shall be changed. 53 For this perishable body must put on the imperishable, and this mortal body must put on immortality. 54 When the perishable puts on the imperishable, and the mortal puts on immortality, then shall come to pass the saying that is written: “Death is swallowed up in victory.” 55 “O death, where is your victory? O death, where is your sting?” 56 The sting of death is sin, and the power of sin is the law. 57 But thanks be to God, who gives us the victory through our Lord Jesus Christ.

This victory over death by our resurrection and rebirth is obtained through the person of Jesus the firstfruit of the new creation:

1 Corinthians 15:20-23 20 But in fact, Christ has been raised from the dead, the firstfruits of those who have fallen asleep. 21 For as by a man came death, by a man has come also the resurrection of the dead. 22 For as in Adam all die, so also in Christ shall all be made alive. 23 But each in his own order: Christ the firstfruits, then at his coming those who belong to Christ.

Jesus has become to us through the new creation the new Adam,

1 Corinthians 15:45-49 45 Thus it is written, “The first man Adam became a living being”; the last Adam became a life-giving spirit. 46 But it is not the spiritual that is first but the natural, and then the spiritual. 47 The first man was from the earth, a man of dust; the second man is from heaven. 48 As was the man of dust, so also are those who are of the dust, and as is the man of heaven, so also are those who are of heaven. 49 Just as we have borne the image of the man of dust, we shall also bear the image of the man of heaven.

And because Jesus is the new creation Adam, we have a new creation genealogy that begins with Christ as the root.

Hebrews makes it clear that this benefit not only applies to New Testament Christians but that also the salvation of the Old Testament saints (including David) similarly rest on the new life in Jesus. Notice how the Hebrew writer speaks of the Old Testament saints this way:

Hebrews 11:39-40 39 And all these, though commended through their faith, did not receive what was promised, 40 since God had provided something better for us, that apart from us they should not be made perfect.

So they are made perfect the same way that we are, in the resurrection, in order to inherit the promises of the kingdom.

Therefore, the root of their new birth in the new creation, as resurrected saints, is the reborn, resurrected, & recreated Jesus himself and, in that way, Jesus can be the root of his ancestors he does not predate except by predating them and us all in the New Creation as God's firstborn.

How can Jesus be both root and offspring of David if he was only a man and did not pre-exist his incarnation?

This is an important question. It deserves to be answered from biblical facts which God has abundantly provided - there is no need for assumptions. Before answering, we do need to rehearse the solid foundation provided regarding Jesus.

He will be great and will be called the Son of the Most High; and the Lord God will give him the throne of his father David; 33and he will reign over the house of Jacob forever, and his kingdom will have no end.” 34But Mary said to the angel, “How will this be, since I am a virgin?” 35The angel answered and said to her, "The Holy Spirit will come upon you, and the power of the Most High will overshadow you; for that reason also the holy child will be called the Son of God". Luke 1:32-5

• Is Jesus mentioned as a living person before his conception? No. Should we assume he does pre-exist based on invented stories appearing 100's of years after the Apostles which none of God's inspired writers affirm? The Gospels tell us when and how Jesus arrived, are consistently unambiguous and affirmed by several biblical authors. We either believe them at face value or choose alternate sources.

• We have several prophecies recorded for us of the 'one to come'. Abraham, 'saw Jesus' day and was glad', John 8:56. David called Jesus his Lord and understood that God would place his descendant on his throne (2Samuel 7:12 13,16, Ps 132:11, 89:3-4). Even prophesying his Lord's life and death Ps 22 - even though he could not grasp the significance of his very specific details.

God of armies, do turn back; Look down from heaven and see, and take care of this vine, 15The shoot which Your right hand has planted, And of the son whom You have strengthened for Yourself. 16It is burned with fire, it is cut down; They perish from the rebuke of Your face. 17Let Your hand be upon the man of Your right hand, Upon the son of man whom You made strong for Yourself. Ps 80:14-17

• A pointer to Jesus ascending to God's right hand - the man God was going to strengthen to do the job He sent him to do - be the saviour of the world and the "Lord" David wrote about, Ps 110.
• Abraham was also advised his descendant would bless the world. Paul also understood where Jesus came from, Gal 3:16

In your (Abraham) seed all nations of the earth shall be blessed, Gen 22:18

For those who believe Jesus did not pre-exist his incarnation, How can Jesus be the root of David if he is David's offspring?

What incarnation? Where do we read about this fanciful idea? The bible contains no story or prophecy about God becoming a man. All we have is a few proof-texts used to push in that direction when read out of context. Or from doubtful translations that have allowed a bias (not intended by God) to alter the scriptures.

Did you notice this bit from the first quotation?

the Lord God will give him the throne of his father David.

• the Lord God will give him... God is giving to who? To God aka Jesus? Why is God giving the throne to God? Did God somehow deserve this reward?
• the throne of his father David. Seems pretty straight forward. God doesn't have a father - certainly not David. But Jesus does have David as a great, great, great... grandfather. One of the gospels affirms this.
• Traditionally, Jesus is God, but this concept doesn't sensibly fit with the clear details provided.

Now how is Jesus the root of David?

God has planned for Jesus to be Saviour from the foundation of the world. Did he already exist back then? No, he was born ~4BC. He is the 'logos became flesh' - same event explained in the gospels. Jesus' name is his identifier. Just as David and Moses and all the other prominent figures had names that identified them and we can trace them through the pages of the bible. Jesus is not mentioned until his conception and birth. He was spoken about prophetically, because his time was not yet.

God, after He spoke long ago to the fathers in the prophets ... 2 in these last days has spoken to us in His Son, whom He appointed heir of all things... Heb 1:1-2

But when the fullness of the time came, God sent His son, born of a woman, born under the Law. Gal 4:4

Nothing about God being born out of a woman or a pre-existing Son who is also God or an incarnation - just a simple birth from a virgin God had chosen to birth His Messiah. We can read-in all that other stuff, but why do we need to? There is no need to presume, if God sent His son, he must have pre-existed! Sending His son is making Mary pregnant! As the gospels tell us over and over without any mystery or complexity.

God has planned everything to happen in a way that leads predictably to the ultimate 'end game'. He has put in place a plan for man's salvation and it is working perfectly. Jesus is the cornerstone of that plan.

you are no longer strangers and foreigners, but you are fellow citizens with the saints, and are of God’s household, 20having been built on the foundation of the apostles and prophets, Christ Jesus himself being the cornerstone, 21in whom the whole building, being fitted together, is growing into a holy temple in the Lord, 22in whom you also are being built together into a dwelling of God in the Spirit. Eph 2:19-

Jesus, the one planned to come, needed a beginning. Not just a birth, but a heritage, a lineage of grand proportions, of righteous men and women who faithfully served God in very demanding situations. God planned for Jesse to be the father of David whose throne Jesus would make endure forever.

1Then a shoot will spring from the stem of Jesse, And a Branch from his roots will bear fruit. 2The Spirit of the LORD will rest on him, The spirit of wisdom and understanding, The spirit of counsel and strength, The spirit of knowledge and the fear of the LORD. 3And he will delight in the fear of the LORD, And he will not judge by what his eyes see, Nor make decisions by what his ears hear; 4But with righteousness he will judge the poor, And decide with fairness for the humble of the earth; And he will strike the earth with the rod of his mouth, And with the breath of his lips he will slay the wicked. 5Also righteousness will be the belt around his hips, And faithfulness the belt around his waist. Isaiah 11:1-5

Being the root of David is the manner by which God had prepared for Jesus' lineage. This holy stock of humanity from which Jesus came. These men and women were set aside for God's purposes - they are the ROOT from which came God's son Jesus. The prophecy of Jesus coming is self-fulfilling - it has it's root in all those God had prepared until Jesus was born.

Then on that day the nations will resort to the root of Jesse, Is 11:10

This is simply a case of God knowing the beginning from the end - Jesus is reason for all that would complete events according to God's sovereign plan. Jesus, the chief cornerstone was planned before the world began.

Jesus is the reason for Jesse! For David and everyone that was part of his story.

for in him all things were created, both in the heavens and on earth, visible and invisible, whether thrones, or dominions, or rulers, or authorities—all things have been created through him and for him Col 1:16

Truly in Jesus all things are held together. Without his saving sacrifice and exalted ascension, nothing would have any meaning. No one would know true life. God's creation would simply cease to exist - it would have no purpose.

Jesus is the root from which all creation flourishes. Not because he made anything, but because he has redeemed everything from evil and sin and death - just as God planned he would. Just as God planned who he would come from and when.

I Jesus have sent mine angel to testify unto you these things in the churches. I am the root and the offspring of David, and the bright and morning star. - Rev 22:16

We read this and we think of Jesus saying, 'before Abraham was, I am'. It all makes perfect sense without reading in anything or jumping to false conclusions. Jesus was before Abraham as being the key, the cornerstone, the reason, the purpose for all things. Abraham was a much smaller part of that plan - of course Jesus was before in the sense of priority and importance - no pre-existence required!

He raised up David to be their king, concerning whom He also testified and said, ‘I have found David, the son of Jesse, a man after My heart, who will do all My will.’ 23From the descendants of this man, according to promise, God has brought to Israel a Saviour, Jesus. Acts 13:22-3