St. Thomas Aquinas
“The washing of the hands is done in the celebration of mass out of reverence for this sacrament; and this for two reasons: first, because we are not wont to handle precious objects except the hands be washed; hence it seems indecent for anyone to approach so great a sacrament with hands that are, even literally, unclean. Secondly, on account of its signification, because, as Dionysius says (Eccl. Hier. iii), the washing of the extremities of the limbs denotes cleansing from even the smallest sins, according to Jn. 13:10: ‘He that is washed needeth not but to wash his feet.’ And such cleansing is required of him who approaches this sacrament; and this is denoted by the confession which is made before the ‘Introit’ of the mass.”
Source: St. Thomas Aquinas, Summa Theologica