Strangely, being served by one who is greater can be as uncomfortable and humbling as being the greatest, yet serving those who are your junior. Peter probably thought: "Lord, I should be washing your feet, not the other way around." Recognizing Jesus as the Messiah and King they'd waited for all their lives, it would be very humbling to have Jesus kneel down and wash his feet.
This is an amazing thing. Both sides are humbled who approach this act as sinners saved by grace, The one who serves recognizes the grace of God at work in their life and is humbled. The recipient is also humbled by having his superior serve him, all the while realizing that he is not worthy to be served in this manner.
This is an important teaching. The greatest shall be as the least of all. This is one of the most important principles of the Kingdom of God. Those who refuse to be served, those who expect to be served, or those who lord over others, or those who refuse to serve others are not part of the Kingdom of God.