When we look again at Jesus' humble act of foot washing [at the Last Supper], we see why the disciples were unable to immediately grasp the significance of the act. Jesus lowered Himself into the position of a lowly slave, He served like a slave, He washed the disciples' feet like a lowest-of-the-low slave, because ultimately He was preparing to die the dehumanizing death of a slave. In essence this is the connection made in Philippians 2:5-8.
As He washed out dirt from between the disciples' toes, Jesus performed a parable of the cross. The disciples could not see the symbolic anticipation, not here, not now. The full explanation for why Jesus washed their feet would only become clear after the substitutionary atonement of the Savior on Good Friday. Then they would look back and understand the act of deep humility in the cross that brought us a once-for-all, head-to-toe, cleansing from our sin.
-- Tony Reinke in an article entitled “The Creator On His Knees”