Del maiale non si butta niente means quite simply that one does not (or at least should not) throw away anything from a pig. As Italy has become richer over the decades, the Italians have taken on our American habit of eating only selected, choice parts of animals, including hogs. Italians have only too gladly given up on things like sanguinaccio, an unspeakable (although I just spoke of it, forgive me) pudding made from pig's blood.
Actually, as part of the sustainability movement (of which I approve), there is a tendency to try to utilize all of the animal, going by the name of whole animal eating. This is going to be a hard sell, as offal is awful, and you probably need to be starving to eat it. But one could compromise and eat more cuts from the animal, if not all.
Does this saying relate to our English-language expression "to go the whole hog?" It would seem so from the meaning (to do something thoroughly), but the origins of the phrase are shrouded in mystery, like so many things.
(The picture shows the famed and beloved maxi-pig Empress of Blandings, from the highly entertaining P.G. Wodehouse)