So I think others have already pointed to numerous meanings around "humility" and standard ritual bathing practices so I will focus on another area. Specifically What "feet" symbolize in the biblical prophetic imagery and how this also related to the symbolism of Israels temple/tabernacle rituals.
Feet in the bible is actually pretty straight forward symbol. The earth itself (and sometimes more specifically the temple) is described repeatidly as being "Gods footstool". Its where God places his feet.
There are many scriptural quotes which shows this but I will start with the most obvious one from Ezekiel 43:7 when God is describing the temple and its "cleansing".
Son of man, this is the place of my throne and the place of the
soles of my feet, where I will dwell in the midst of the people of
Israel forever. And the house of Israel shall no more defile my holy
name, neither they, nor their kings, by their whoring and by the dead
bodies2 of their kings at their high places,
But there are many others.
Isaiah 66:1 This is what the LORD says: “Heaven is My throne, and
earth is My footstool.
Matthew 5:34 But I tell you, do not swear an oath at all: either by
heaven, for it is God’s throne; or by the earth, for it is his
Chronicles 28:2 Then King David rose to his feet and said: “Hear me,
my brothers and my people. I had it in my heart to build a house of
rest for the ark of the covenant of the Lord and for the footstool of
our God, and I made preparations for building
Psalm 32:7 “Let us go to his dwelling place; let us worship at his
In addition to these very direct and overt references to the earth/temple and Gods feet there is the deeper and more elaborate prophetic imagery around this. It is described in the temple construction given by God to Moses and the image of the "Son of Man" and also the 4 living creatures. These are described as having "burnished bronze feet". See Revelation 1:15, Daniel 10:6 and also Ezekiel 1.
His feet were like polished bronze refined in a furnace
When Moses created the Tabernacle in the wilderness by the image of the pattern he had seen from heaven it was constructed with different "areas". Each area was representative of something within Gods kingdom. There was an "Outer Court" where the Brazen bronze Altar used for animal sacrific was located and the polished "Bronze Laver". (Hebrew: כיור נחשת kîyōr nəḥōšeṯ). It was seperated from the "tent of meeting" by a curtain. Priests were required to wash their feet and sometimes hands in the bronze basin before completing the sacrifice of atonement and entering into the "inner sanctuary" and the "Holy of Holies" within the "tent of meeting".
See an image which shows the layout of the tabernacle here -> https://i.stack.imgur.com/yilEo.jpg
Exodus 30:18 “You shall also make a basin of bronze, with its stand
of bronze, for washing. You shall put it between the tent of meeting
and the altar, and you shall put water in it, Aaron and his sons are
to wash their hands and feet with water from it. Whenever they
enter the tent of meeting, they shall wash with water so that they
will not die.
They made the bronze basin and its bronze stand from the mirrors of
the women who served at the entrance to the tent of meeting.
Many theologians associate the symbolism of the outer court and "bronze" with "earthly blood sacrifice" and atonement. It obtains a certain degree of purification required for humanity to then enter into the tent of meeting and inner 'Holy of Holys' to commune with God. That inner section of the temple is covered in Gold which most believe represented the heavenly abode of God. The heavenly figures with burnished bronze feet pointed to the earthly ritual purification and sacrifice required for the place God would "place his feet". His Footstool.
In this respect at the last supper we see Jesus right before he enters into his passion. His earthly sacrifice on the "brazen alter" of the cross as the "lamb of God". He sits with his 12 disciples who represent Israels 12 tribes. His washing of their feet "symbolically" represents the cleansing of Israel / Temple and the Earth. It is the washing of "Gods feet". "Cleansing his earthly temple". His footstool. This then allows for the curtain that separated the "brazen outer court" from the Golden "inner sanctum" to be opened.
Then Jesus had cried out again in a loud voice, He yielded up His
spirit. At that moment the veil of the temple was torn in two from top
In short summary Jesus only washes the "feet" because its symbolic of washing of Israel / The temple / the earth. The rest of Gods abode and kingdom and body is already clean and does not require washing. Also this shows the "humility" of God. That the almighty - the one who sits on the throne as King with supreme authority - who rightfully should be "served" by those at his feet - has instead lowered himself down and come to serve and cleanse them.