Jesus set an example by washing the disciples’ feet. In those days it was necessary to have the servant wash your guests’ feet clean from all the horse and donkey poo they would get as a normal part of traveling from one place to another. The washing meant that the guest would not mess up your clean house. Our house must be kept clean. Imagine reclining at table in the ancient Roman fashion and smelling your neighbor’s dog-poo feet.
The significance is that Jesus cleaned the feet: his actions made them worthy of entering the house. Peter had to allow Jesus to wash his feet; he was not wanting the Lord to lower himself to the task. Similarly we have to be willing to allow our feet to be washed, to be helped to be better through others’ advice, aid, etc. And we must wash our brothers’ feet: be willing to help them. But before everything we must allow Jesus to wash out feet. If we go to someone’s house (and heaven is Jesus’ house) we must respect the wishes of the owner; it’s only polite.