At the very beginning of the play, there is a quote, in old Italian, from an old author named Dante. Dante is famous for writing "The Divine Comedy," or, "The Inferno." This tale is about the various different levels of Hell. We are taken on a journey through the different levels; each level is for different types of crimes, the lowest level being reserved for traitors to God. Anyway, the quote that Eliot references refers to a man in Hell who had tried to gain forgiveness for a crime before he ever even committed it. For a word-for-word interpretation, see my link below; eNotes provides a great interpretation.
Eliot probably put that quote because the entire poem is Prufrock wanting to know the answer to his question without having to ask it. He is too afraid to ask it if it will be a negative; he hopes that he can be assured of a positive answer before taking the plunge and asking the woman either to marry him, or before revealing his feelings for her. He doesn't want to ask if he is going to get rejected, just like the man in Hell didn't want to commit the crime if it meant he would be punished and unforgiven. He wanted to be able to commit the crime, AND not be punished, just as Prufrock wants a positive answer before he asks the question. It's an interesting annotation to put at the beginning of the poem, and helps us to gain greater insight into Prufrock's state of mind as he ponders his "overwhemling question." I hope that helps; good luck!
Another perspective on the Italian epigraph in which the man in Hell tells a visitor that he would never tell his story if there were a chance that it would get back to living ears is that in his neurosis Prufrock, like the man in Hell, retreats into an interior monologue: one that cannot be heard by "living ears." Thus, the poem becomes a young man's agonizing over a woman he loves. This exercise in futility suggests the isolation of modern man as Prufrock includes a "you" in the poem which reflects his separation from the world.