What does John 13:6 mean?

Jesus is performing a grimy, lowly, humble chore: washing the feet of the disciples (John 13:4–5). This is something only done by servants and slaves—never by a master for his own subjects. This is why Peter initially responds with an expression of disbelief. The disciples have shown an interest in power and prestige (Mark 9:34; Luke 9:54), so Peter wouldn't be appalled to think "someone" would wash his feet. What he objects to is that Jesus would do it. His question is meant along the lines of, "why are You, Jesus, doing this, instead of someone else?"

The answer given later is that humble servanthood is something followers of Christ are obligated to pursue. Jesus is unquestionably the Master, and if He will serve others, those who claim His name must do so, as well (John 13:12–16). In the short term, Peter will misunderstand Jesus' intentions, and ask to be made entirely clean. This provides an opportunity to explain the difference between being spiritually "bathed," once-and-for-all, versus being "washed" on a regular basis (John 13:9–10).