John 13:10 is an important verse for our spiritual lives. Jesus used a physical action to illustrate a spiritual truth. There are 3 Greek words that bring out the spiritual truth behind the foot washing.
The first word is λελουμενος ("the one who has been bathed"). This word referred to the washing of the whole body (Heb. 10:22; Acts 9:37). It was used in the literature of the day for bathing the body in the bathing room. An interesting use of the word is found in 2 Peter 2:22 which is best translated "the sow that washes itself by wallowing in the mire" (Moulton & Milligan, Vocabulary, p. 381). Some people bathe themselves in the muck of sin but we have been bathed in the pure water of Jesus. The form of the verb refers to action that occurred in the past with ongoing results in the present.
The second word is νιψασθαι ("to wash"). This word refers to washing a part of the body like hands or feet (Liddell & Scott, Lexicon, cf. Mark 7:3; Mt. 6:17). The verb is an Aorist Middle Infinitive which is probably best understood as a permissive middle - "to let his feet be washed" (S.Lewis Johnson, private notes). This washing is not a washing done to us in the past but a washing we must allow whenever we get spiritually dirty in the present.
The third word is καθαρος ("clean"). It is used in tandem with ολος ("wholly or completely"). It meant to be pure, innocent or free from offence (Liddell & Scott, Lexicon). The one who has been bathed by Jesus (saved) is wholly pure. Jesus continues by saying "You are pure ones, but not all." Obviously, He is referring to Judas as the exception to purity in their midst even though Jesus has probably washed his feet physically. All who have been spiritually bathed by Jesus (regeneration) are pure on an ongoing basis (present tense) even though they get dirty in life.
The illustration refers to a person who bathes at home but gets dust on his feet walking to the banquet. Such a person only needs to wash his feet to enjoy the banquet. We get dirty as we walk through life and we need to let Jesus wash our feet spiritually to enjoy the intimacy of dinner with Him.