The common explanation for this is that it teaches us to learn humility by doing good for others, by doing acts of service or kindness for our brethren. This is certainly a good lesson that we can take from Christ's example, but we can perhaps derive another from it.
In John's account, what did Jesus suggest that the washing of feet symbolized? He tells Peter that the washing of his feet symbolizes forgiveness of his sin to return him to a "clean" relationship with God. It is only logical to deduce that God expects nothing less from us in response to the sins of our brethren. In the section of the Sermon on the Mount on prayer, Jesus says: "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).
No doubt, God puts a very great emphasis on our relationships since our lives are to reflect His character. If we have begun to "put on Christ" (Galatians 3:27), would we be a good example of His love for us if we held grudges, hated our brother, or would not forgive another? Obviously, no. Putting on Christ demands that we "put off" these carnal destroyers of relationships and replace them with Christian virtues.
Peter asks Christ, "Lord, how often shall my brother sin against me, and I forgive him? Up to seven times?" (Matthew 18:21). Christ's answer should give us a clue to how He feels about this issue. Peter had ventured a number he thought would be sufficient to establish his forbearance. Christ, though, pulls out all the stops, telling him that there is no set limit: "I do not say to you, up to seven times, but up to seventy times seven" (verse 22). We are indeed fortunate and can be thankful that same unlimited forgiveness applies to us when we need God's mercy.
The following verses, Matthew 18:23-35, is the Parable of the Unforgiving Servant. The servant was deeply in debt to his master, and when he sought relief, his master forgave him his gargantuan debt. Then the tables turn. Another man owed him a small amount and could not repay it. Instead of following his master's example, the servant forgot the mercy he had just received and had the man thrown into prison!
Verses 34-35 sum up the story: "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 will do to you if each of you, from his heart, does not forgive his brother his trespasses." The language Christ uses leaves little room for exclusions. He Himself, in the agony of crucifixion, says without reservation, "Father, forgive them, for they do not know what they do" (Luke 23:34). His plea applies, not only to those who cried out for His death and nailed Him to the stake, but to all, past and future, who would be just as responsible as they were and need God's forgiveness. That includes everybody.