This ceremony is known as Urchatz, "Washing of the Hands".  This is a symbolic act of purification, which precedes participation in the religious service.  This comes from Psalm 24:3-4:
Who may ascent the hill of the Lord?  Who may stand in his holy place?  He who has clean hands and a pure heart.  Psalm 24:3-4 
    Yeshua taught us that the sign of a pure heart is the heart of a  servant when he washed the feet of his disciples.   

    After that, he poured water into a basin and began to wash his disciples' feet, drying them with the towel that was wrapped around him.  John 13:5 

     When he had finished washing their feet, he put on his clothes and returned to his place.  "Do you understand what I have done for you?"  he asked them.  "You call me 'Teacher' and 'Lord,' and rightly so, for that is what I am.  Now that I, your Lord and Teacher, have washed your feet, you also should wash one another's feet.  I have set you an example that you should do as I have done for you.  I tell you the truth, no servant is greater than his master, nor is a messenger greater than the one who sent him.  Now that you know these things, you will be blessed if you do them." John 13:12-17