After Jesus washed the disciples' feet, he revealed (as promised to Peter) the purpose of washing feet.
Jesus first noted that the disciples referred to Jesus as Lord and Master, which they said well (13:13). The title of "Lord" and "Master" indicates that Jesus has power and authority and is basically their guide. But with this power and authority, Jesus came down to serve the disciples via washing their feet.
Jesus also called the disciples to wash others' feet, with Jesus as example (13:14-15). He sets an example by washing feet to demonstrate what service is like. Jesus also talked about how a servant is no greater than a master (13:16). Thus, when serving, we aren't called to elevate ourselves but focus on God as Lord.
Finally, Jesus blessed the disciples when they do such things (13:17). He blesses the disciples because serving is important and impactful. Even today, Jesus calls us to serve with humility.