Epigraphs are like little appetizers to the great entrée of a story. They illuminate important aspects of the story, and they get us headed in the right direction.
Book 3: Multitudinis imperitae non formido judicia; meis tamen, rogo, parcant opusculis—in quibus fuit propositi semper, a jocis ad seria, a seriis vicissim ad jocos transire.
"I do not fear the opinions of the ignorant crowd; nevertheless, I ask they spare my little work, in which it has always been my purpose to pass from humour to seriousness and from seriousness back to humour."
Sterne adapted this epigraph from Rabelais, who borrowed it from the Bishop of Chartres, who lived in the 12th century. Those names don't really matter. Tristram's just messing with an old tradition that requires authors to make humble-pie. See, over-the-top epigraphs like this are pretty standard in old books. The author directly addresses the audience, asking that readers approach the work with an open mind. What's different about this epigraph is that Sterne adds the last bit about how his plan is to move from being funny to being serious. He's priming the reader to expect the book to be melancholy and lighthearted at the same time—a tragicomedy with a twist.
Book 5: Dixero si quid fortè jocosius, hoc mihi juris/ Cum venia dabis.—Horace
"If in my words I am too free, perchance to light, this bit of liberty you will indulgently grant me."
Si quis calumnietur levius ese quam decet theologum, aut mordacius quam deceat Christianum—non Ego, sed Democritus dixit.—Erasmus
"If I have overshot myself in this which hath been hitherto said, or that it is, which I am sure some will object, too fantastical, too light and comical for a Divine, too satirical for one of my profession, I will presume to answer with Erasmus, in like case, 'Tis not I, but Democritus"
These two epigraphs come from Burton's Anatomy of Melancholy, a major book about depression from 1621, well before the days of Prozac and Abilify.
It's evidently one of Sterne's favorite books, since parts of Book 5 and 6 are practically plagiarized from it. Like the previous epigraph, they ask the reader to approve of his work and give the reader some hints at the tone of the book. Sterne was a priest in the days before church services featured slideshows and drum sets. He may have thought that (like Yorick) people would think the book wasn't exactly an appropriate side project. Drop in a vaguely apologetic epigraph, though, and we're doing business.
Book 7: Non enim excursus hic ejus, sed opus ipsum est. (This is not a digression, but his main work.) Pliny the Younger, Epistles.
Book 7 launches into the European journey, taking us far away from Shandy Hall. Not to worry, Sterne tells us in this epigraph, because the digression is actually the most important part.
You know who talks about digressions a lot? Tristram. It's worth thinking about where Sterne ends and Tristram begins, and epigraphs—which are a kind of what scholars call paratexts, the framework for a book—are a good place to start.
Book 9: Si quid urbaniusculè lusum a nobis, per/ Musas et Charitas et omnium poetarum/ Numina, Oro te, ne me malè capias.
"Though you might prefer a somewhat more polite amusement, by the Muses and Charities and the grace of all poets, do not think badly of me."
Sterne picks up his Burton again, although this epigraph is a combination of a couple of lines rather than a direct quote. He asks the reader to accept that the story he's about to tell is a little bawdy (not "polite")—after all, this is the climax (or not) of Toby's courtship.
These epigraphs have one thing in common: they force the reader to read the book in a certain way. It's almost as though Sterne doesn't trust the reader to figure it out on his own—like he's writing for a bunch of clueless Tobys.