The priest does not wash his hands because they are dirty, but to signify a spiritual meaning.
When attending a Catholic Mass, you may notice that the priest washes his hands immediately before reciting the Eucharistic prayer at the altar.
At first glance it may seem as though the priest is washing his hands because of dirt, but if that were the only reason, why not wash his hands before Mass?
The reason is spiritual, as St. Cyril of Jerusalem notes in his Catechetical Lectures.
You have seen then the Deacon who gives to the Priest water to wash, and to the Presbyters who stand round God’s altar. He gave it not at all because of bodily defilement; it is not that; for we did not enter the Church at first with defiled bodies. But the washing of hands is a symbol that you ought to be pure from all sinful and unlawful deeds; for since the hands are a symbol of action, by washing them, it is evident, we represent the purity and blamelessness of our conduct. Did you not hear the blessed David opening this very mystery, and saying, I will wash my hands in innocency, and so will compass Your Altar, O Lord ? The washing therefore of hands is a symbol of immunity from sin.
The priest says a quiet prayer at this point in the liturgy that reinforces this reality.
It is a simple gesture, but one that has profound spiritual meaning.
The ritual is meant to remind the priest that he is called to live purely and devoutly, avoiding all sin.
Everything in the liturgy has a meaning, though it may not be clearly evident.