Before we look at John, who records the foot washing, we must look at Luke who tells of what the apostles were discussing.
Luke 22:23-30 And they began to discuss among themselves which one of them it might be who was going to do this thing. And there arose also a dispute among them as to which one of them was regarded to be greatest. And He said to them, "The kings of the Gentiles lord it over them; and those who have authority over them are called 'Benefactors.' "But not so with you, but let him who is the greatest among you become as the youngest, and the leader as the servant. "For who is greater, the one who reclines at the table, or the one who serves? Is it not the one who reclines at the table? But I am among you as the one who serves. "And you are those who have stood by Me in My trials; and just as My Father has granted Me a kingdom, I grant you 30 that you may eat and drink at My table in My kingdom, and you will sit on thrones judging the twelve tribes of Israel.
Jesus here is telling the Apostles that they must be men who are servants, just as Jesus is a servant. Notice also that He talks about them having a leadership role when He compares them with Kings and says how they will sit on thrones and act as judges. He is directly speaking to them, not just as Christians, but in their office as Apostles.
Jesus in the Gospel of John shows the same type of thing in Chapter 13:3ff:
Jesus, knowing that the Father had given all things into His hands, and that He had come forth from God, and was going back to God,
Jesus came from God as an Authority, the Authority, and when His mission was completed He would return to the Father in Heaven. The Apostles in their ministry did this very thing. Jesus sent them out to preach the good news and cast out demons and they returned to give a report. Jesus Himself was an Apostle, St. Paul tells us:
Hebrews 3:1-2 consider Jesus, the Apostle and High Priest of our confession. He was faithful to [God] who appointed Him.
I am just trying to establish here that the context is apostleship.
Continuing on and notice the detail that St. John gives: Listen to verse 4 and 5: Jesus rose from supper, and laid aside His garments; and taking a towel, He girded Himself about. Then He poured water into the basin, and began to wash the disciples' feet, and to wipe them with the towel with which He was girded.
I am going to include verse 12 here and you will see why in a minute.
And so when He had washed their feet, and taken His garments, and reclined at the table again,
Notice again what Jesus does. He takes off His garments, puts on a towel and washes their feet with water and wipes them with the towel He has put on. When He has finished, He puts His own garments back on again.
Philippians 2:6-11 who, although He existed in the form of God, did not regard equality with God a thing to be grasped, but emptied Himself, taking the form of a bond-servant, and being made in the likeness of men. And being found in appearance as a man, He humbled Himself by becoming obedient to the point of death, even death on a cross. Wherefore God also hath highly exalted him, and given him a name which is above every name: That at the name of Jesus every knee should bow, of things in heaven, and things in earth, and things under the earth; And that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.
So Jesus is exalted but then humbles Himself and then is exalted again. We have spoken before on this show how garments in the Bible represent glory. Jesus hides that glory and becomes a man and later resumes that glory. This is the symbolism of his garments. He takes off his garment and puts on that of a servant. Jesus then takes the dirt from the Apostles and puts it on Himself. This points us to the Cross, where Jesus would suffer death on our behalf, only to rise again in glory on the third day.
Now while that is an allegorical interpretation of what has happened here, Jesus is doing more than just washing their feet. This is part of their ordination as priests.
Exodus 40:12, 13, 31, 32 "Then you shall bring Aaron and his sons to the doorway of the tent of meeting and wash them with water. "And you shall put the holy garments on Aaron and anoint him and consecrate him, that he may minister as a priest to Me. …And from it Moses and Aaron and his sons washed their hands and their feet. When they entered the tent of meeting, and when they approached the altar, they washed, just as the LORD had commanded Moses.
What Jesus is doing is just what Moses was doing with the priests in his day. This is a part of their ordination. This is what we commemorate at the foot washing. We are reenacting the 12 Apostles ordination to the priesthood. It is for this reason of representing the original 12 Apostles that the church says that the 12 who are getting their feet washed need to be 12 men.
John continues in chapter 13 verse 6:
And so He came to Simon Peter. He said to Him, "Lord, do You wash my feet?" Jesus answered and said to him, "What I do you do not realize now, but you shall understand hereafter." Peter said to Him, "Never shall You wash my feet!" Jesus answered him, "If I do not wash you, you have no part with Me." Simon Peter said to Him, "Lord, not my feet only, but also my hands and my head."
What Jesus says here points us back to the larger context of Jesus dying for us. It is as if Jesus is saying, “If you don’t let me die for you, Peter, then you won’t get Heaven.” And Peter answers with, “ wash me all over then.”
Jesus said to him, "He who has bathed needs only to wash his feet, but is completely clean; and you are clean, but not all of you." For He knew the one who was betraying Him; for this reason He said, "Not all of you are clean." And so when He had washed their feet, and taken His garments, and reclined at the table again, He said to them, "Do you know what I have done to you? You call Me Teacher and Lord; and you are right, for so I am. If I then, the Lord and the Teacher, washed your feet, you also ought to wash one another's feet. For I gave you an example that you also should do as I did to you.
Now recall back to Luke what the Apostles were discussing. They were talking about who will be the greatest. See what Jesus replies here in John:
"Truly, truly, I say to you, a slave is not greater than his master; neither is one who is sent greater than the one who sent him. "If you know these things, you are blessed if you do them.
So if God can get on His hands and knees and serve us, what then should the Apostles be like? Are they better than the master? No.
Now verse 20 keeps this whole event in kind of bookends:
"Truly, truly, I say to you, he who receives whomever I send receives Me; and he who receives Me receives Him who sent Me."
This sounds similar to where Jesus in Luke says:
Luke 10:16 "The one who listens to you listens to Me, and the one who rejects you rejects Me; and he who rejects Me rejects the One who sent Me."
My point is that the context of the foot-washing is the ordination of the Apostles at the Last Supper and their sending forth, like the Son of God, to be servants to the world.
Here is the document that says that it should be men who get their feet washed.
The Preparation And Celebration Of The Easter Feasts
Congregation for Divine Worship
51. The washing of the feet of chosen men (Latin 'viri' - men) which, according to tradition, is performed on this day, represents the service and charity of Christ, who came "not to be served, but to serve. This tradition should be maintained, and its proper significance explained.