Why does the Lord's Prayer instruct us to ask God to forgive us "as we forgive others"?

In addition to the answer given by bruised reed the reason we have a hard time forgiving those who have wronged us lies in pride, (which God hates) for some reason we tend to consider our sins as lesser than those who sinned either against us, or for that matter against God. For some reason we cannot accept that there is no difference between a white lie and worshipping Satan.

God only has one classification of sin, and that being all are worthy of death.

Romans 3:23 for all have sinned and fall short of the glory of God,

We can only expect to receive as forgiveness for what we are willing to forgive in others, after all if a sin is not forgivable to all how can it be forgivable to any?

And since God can and will forgive them for their sins when they pray and repent, do you have the right to continue to hold them accountable?

While it is true that we do not have the ability to forgive sin Jesus does and if we have him in our hearts, we can ask him to give us the power to forgive since:

John 14:12 through 14 "Most assuredly, I say to you, he who believes in Me, the works that I do he will do also; and greater works than these he will do, because I go to My Father. 13 And whatever you ask in My name, that I will do, that the Father may be glorified in the Son. 14 If you ask anything in My name, I will do it.

And Jesus did in fact forgive sins while on Earth in a human body.

Matthew 9:2 Then behold, they brought to Him a paralytic lying on a bed. When Jesus saw their faith, He said to the paralytic, "Son, be of good cheer; your sins are forgiven you."

Luke 7:47 and 48 Therefore I say to you, her sins, which are many, are forgiven, for she loved much. But to whom little is forgiven, the same loves little." 48 Then He said to her, "Your sins are forgiven."

We have to remember that in his human form Jesus was tempted in every way we are and that includes the temptation to not forgive those who sin against us, and yet while dying on the cross said:

Luke 23:34 Then Jesus said, "Father, forgive them, for they do not know what they do." And they divided His garments and cast lots.

If we expect Grace we should ask of God in Jesus name and he will give us the grace to forgive others.

Hope this helps.

Because God ties our forgiveness with our own behavior: if we don’t forgive, then we won’t be forgiven. You quoted the Lord's Prayer, but neglected to read the next two verses:

Matthew 6:14-15

For if you forgive others their trespasses, your heavenly Father will also forgive you, but if you do not forgive others their trespasses, neither will your Father forgive your trespasses.

http://www.blueletterbible.org/Bible.cfm?b=Mat&c=6&t=ESV#s=935014

How much do we want to be forgiven? How much mercy do we want to receive from God? God will give to us the same measure we extend to others (Matthew 7:2).

James 2:13 sums it up well, “For judgment is without mercy to the one who has shown no mercy. Mercy triumphs over judgment.”

God teaches us to ask for forgiveness 'as we forgive others' for many good theological reasons. I want to focus on one that is as practical as it is theological.

Do sinners have any idea what it has cost God to forgive them? No, they don't. This price they will ever learn more of as they take a good and honest look at the cross.

I can see the divine 'genius' in teaching us to pray for forgiveness in this way because it helps us understand like nothing else what forgiveness costs. Yes, we may never fully comprehend the full magnitude of God's forgiveness for us, but this way we all of a sudden have some 'skin in the game'.

As we then forgive others that have wronged us, we get a sense of the giving end of forgiveness and not merely receiving it. It is so easy for us to make God's forgiveness cheap when we start living in sin and simply reason that tonight, before I go to bed, I will take out 'God's forgiveness brush' and clean up the mess I have made today.

I hope you can see how this kind of forgiveness is meaningless and certainly not what God intended. He teaches us to pray this way so that we don't abuse His forgiveness but 'keep it real' by requiring us to forgive others as we are forgiven and not merely because we are forgiven.

Our forgiveness of others is the only way we can begin to understand His costly forgiveness for us. God may be considered as very rich since He owns the universe, but sin hit God where He was very poor - He only had one Son. We should never get the idea that because God is rich in grace and forgiveness, that His grace and forgiveness did not cost Him something very real and painful.

Because as the reading said at Mass today (!) Unworthy Servants 7 “Will any one of you who has a servant plowing or keeping sheep say to him when he has come in from the field, ‘Come at once and recline at table’? 8 Will he not rather say to him, ‘Prepare supper for me, and dress properly, and serve me while I eat and drink, and afterward you will eat and drink’? 9 Does he thank the servant because he did what was commanded? 10 So you also, when you have done all that you were commanded, say, ‘We are unworthy servants; we have only done what was our duty.’”

AND NOTICE THIS IS ABOUT THOSE WHO HAVE DONE ALL THAT YOU WERE COMMANDED. IS that you?

Jesus's model prayer praises God, shows hope that God's plan will proceed, and asks for personal favours from God. But, as noted, there is one additional phrase attached to it: "as we forgive others".

In this model prayer, there is one and only one requirement placed on petitioners, that they have forgiveness. This condition is the single most important thing God asks.

Much of the Hebrew scriptures is a demonstration of God's character. The rituals and ceremonies are all about God's forgiveness of sins. These physical symbols of forgiveness foreshadowed the spiritual forgiveness that would become available following Jesus's resurrection.

The hope of Christians is to be God's children, Jesus's siblings. And just as Adam was created in God's image, at Christ's return they will become immortal spirit beings as part of the first general resurrection, reborn in God's spiritual image, bearing his perfect character.

It is a Christian's duty to develop a perfect character like Jesus's, like God's, to truly be in God's image.

But it doesn't matter how many good deeds one does, how much one gives to charity, how much one shows love for their fellow man. The true test is whether one can remove all negative feelings towards others from their hearts.

Consider the following scriptures:

For if you forgive men their trespasses, your heavenly Father will also forgive you. But if you do not forgive men their trespasses, neither will your Father forgive your trespasses.
— Matthew 6:14–15

Then Peter came to Him and said, “Lord, how often shall my brother sin against me, and I forgive him? Up to seven times?” Jesus said to him, “I do not say to you, up to seven times, but up to seventy times seven.
— Matthew 18:21–22

[Parable]
Then his master, after he had called him, said to him, ‘You wicked servant! I forgave you all that debt because you begged me. Should you not also have had compassion on your fellow servant, just as I had pity on you?’ “And his master was angry, and delivered him to the torturers until he should pay all that was due to him. “So My heavenly Father also will do to you if each of you, from his heart, does not forgive his brother his trespasses.”
— Matthew 18:32–35

And whenever you stand praying, if you have anything against anyone, forgive him, that your Father in heaven may also forgive you your trespasses. But if you do not forgive, neither will your Father in heaven forgive your trespasses.
— Mark 11:25–26

But I say to you who hear: Love your enemies, do good to those who hate you, bless those who curse you, and pray for those who spitefully use you. To him who strikes you on the one cheek, offer the other also. And from him who takes away your cloak, do not withhold your tunic either. Give to everyone who asks of you. And from him who takes away your goods do not ask them back. And just as you want men to do to you, you also do to them likewise.

But if you love those who love you, what credit is that to you? For even sinners love those who love them. And if you do good to those who do good to you, what credit is that to you? For even sinners do the same. And if you lend to those from whom you hope to receive back, what credit is that to you? For even sinners lend to sinners to receive as much back. But love your enemies, do good, and lend, hoping for nothing in return; and your reward will be great, and you will be sons of the Most High. For He is kind to the unthankful and evil. Therefore be merciful, just as your Father also is merciful.

Judge not, and you shall not be judged. Condemn not, and you shall not be condemned. Forgive, and you will be forgiven. Give, and it will be given to you: good measure, pressed down, shaken together, and running over will be put into your bosom. For with the same measure that you use, it will be measured back to you.
— Luke 6:27–38

Take heed to yourselves. If your brother sins against you, rebuke him; and if he repents, forgive him. And if he sins against you seven times in a day, and seven times in a day returns to you, saying, ‘I repent,’ you shall forgive him.
– Luke 17:3–4

Therefore let it be known to you, brethren, that through this Man is preached to you the forgiveness of sins; and by Him everyone who believes is justified from all things from which you could not be justified by the law of Moses.
— Acts 13:38–39

But if anyone has caused grief, he has not grieved me, but all of you to some extent—not to be too severe. This punishment which was inflicted by the majority is sufficient for such a man, so that, on the contrary, you ought rather to forgive and comfort him, lest perhaps such a one be swallowed up with too much sorrow. Therefore I urge you to reaffirm your love to him. For to this end I also wrote, that I might put you to the test, whether you are obedient in all things. Now whom you forgive anything, I also forgive. For if indeed I have forgiven anything, I have forgiven that one for your sakes in the presence of Christ, lest Satan should take advantage of us; for we are not ignorant of his devices.
— 2 Corinthians 2:5–11

Let all bitterness, wrath, anger, clamor, and evil speaking be put away from you, with all malice. And be kind to one another, tenderhearted, forgiving one another, even as God in Christ forgave you.
— Ephesians 4:31–32

Therefore, as the elect of God, holy and beloved, put on tender mercies, kindness, humility, meekness, longsuffering; bearing with one another, and forgiving one another, if anyone has a complaint against another; even as Christ forgave you, so you also must do. But above all these things put on love, which is the bond of perfection.
— Colossians 3:12–14

Even secular medicine tells us that resentment, hatred, grudges, desire for vengeance, etc. is bad for our physical and mental health. It is not possible to have God's perfect character while retaining these feelings.

Satan has injected so many lies into what is generally taught as Christianity today because he doesn't want people to develop God's character. These anti-christian concepts, such as penance, Purgatory, and Hell, teach the exact opposite of what God wants.

Once someone has experienced remorse for their ways and has repented (actually changed their character), there is no point in anything further. God has forgiven the sin, and what happened before is no longer of any significance. It simply doesn't make sense to punish someone after they have changed.

The rationalization is that somehow they still need to suffer, to make penance, in order to make up for what they did. But there is nothing anyone can do to pay for their own sins; only Jesus's death can do that.

People talk about going to Heaven, perhaps being able to observe the sinners being tortured in Hell for all eternity. That is simply not the way of the God of the Bible. God is not going to torture people for all eternity (or even for a short while); his way is one of love not hate.

This current age is a time of training and learning for God's elect; but most people are not being offered salvation at this time. They will receive their offer of salvation at the end of the Millennium, when the vast majority of mankind will be physically resurrected.

God's desire is that all will be saved, though unfortunately there will be a small number that will refuse to accept his way. These unrepentant sinners will be mercifully destroyed, not tortured or made to suffer.

If one meets Stalin or Hitler or Judas or whoever after the second resurrection, and finds that they now realize the enormity of their sins and have repented, how should one respond? Will it be with joy that a fellow sibling has been saved, or with resentment or worse?

That is the true test of whether one really is a Christian.

why are we taught to ask God only to forgive us in the manner that we forgive others? It seems we would want a much greater level of forgiveness — the kind of forgiveness that only God can give.

Exactly. Christians should strive to be like God.

He who does not love does not know God, for God is love.
— 1 John 4:8

But I say to you, love your enemies, bless those who curse you, do good to those who hate you, and pray for those who spitefully use you and persecute you, that you may be sons of your Father in heaven; for He makes His sun rise on the evil and on the good, and sends rain on the just and on the unjust. For if you love those who love you, what reward have you? … And if you greet your brethren only, what do you do more than others? … Therefore you shall be perfect, just as your Father in heaven is perfect.
— Matthew 5:44–48

Forgiving others is very, very important. Firstly, we emphasize the primary importance of love in the life of a believer:

The goal of this command is love, which comes from a pure heart and a good conscience and a sincere faith. 1 TImothy 1:5 NIV

Compare also the well known Great Commandment. The vital importance of forgiving others is made manifest when we realize that it is utterly impossible to truly love those we harbor unforgiveness towards (certainly from 'a pure heart' or 'with all our heart, mind, soul and strength' or 'as ourselves')- we deceive ourselves if we think so. The gospels are quite clear that the graceless attitude of unforgiveness will prevent us from receiving grace ourselves, in the succeeding verses to the Lord's prayer given in Matthew:

14 For if you forgive other people when they sin against you, your heavenly Father will also forgive you. 15 But if you do not forgive others their sins, your Father will not forgive your sins. - Matthew 6:14-15 NIV

Compare also The Parable of the Unmerciful Servant.

Viewed in human terms, the 'manner' of our forgiveness will not of course match the Lord's in view of His sublime perfection; however, God is fully committed to supply the required grace to enable us to forgive completely and sincerely if we trust him to do so:

His divine power has given us everything we need for a godly life through our knowledge of him who called us by his own glory and goodness. - 2 Peter 1:3 NIV