Epigraphs are like little appetizers to the great main dish of a story. They illuminate important aspects of the story, and they get us headed in the right direction.
"When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him."
—Jonathan Swift, Thoughts on Various Subjects, Moral and Diverting
What's up with the epigraph?
The epigraph is from an essay by the Anglo-Irish satirist and wit Jonathan Swift. The essay, "Thoughts on Various Subjects, Moral and Diverting," isn't actually an essay; it's just a collection of epigrams and clever sayings that was published in 1706. People didn't have the Internet back then, so sometimes they used books for the sort of random disconnected thoughts that today you might put on your Facebook or Tumblr where they belong.
As for what the epigram-cum-epigraph means, well, it means exactly what it says it means. You can tell a true genius by the fact that all the idiots and bounders and knuckleheads are arrayed against him. In short, all the dunces hate a genius.
So who is the genius in this book? Why Ignatius Reilly, of course. Just ask him, and he'll happily tell you so himself. America, he insists, "has no contact with reality. That is only one of the reasons that I have always been forced to exist on the fringes of its society" (5.178). Reilly knows the truth that America is afraid to face; therefore he is beset by dullards, like his mother, who wants him to work, or Mr. Gonzalez, who wants him to work, or Mr. Clyde, who wants him to work, or possibly Myrna, who wants him to have sex.
Does the novel really believe that Reilly is a genius, though? If everyone is against him, it seems like it's because he's a sloth and an insufferable jerk, rather than because he's especially insightful. And for that matter, lots of other people in the novel seem to have the dunces of the world working against them as well. Mrs. Reilly, for instance, certainly has enough troubles. Or what about poor Officer Mancuso staking out that bus bathroom by order of his malevolent sergeant? Everyone, it seems, has a confederacy of dunces in their way.
And if everyone has a confederacy of dunces in their way, then everybody is a genius. And then there can't be any dunces left. Hm. Interesting.
You might think that the genius against whom the dunces rail is not Reilly, but the author, John Kennedy Toole. Toole, after all, was not recognized in his lifetime. A Confederacy of Dunces is much beloved now, but when it was written Toole couldn't find a publisher. The epigraph then, might be seen as a subtle dig at all those publishing houses that couldn't appreciate genius when they saw it.
Except… it probably isn't. Toole had high hopes of publishing his book when he named it, so if he was a genius, he was one who didn't know that the dunces were out to get him. Or at least, he didn't know they were out to get him anymore than they're out to get everybody, genius and dunce alike.
Bonus Feature
Note that there's a kind of second, bonus epigraph after the first epigraph but before the first chapter. Score, right?
This long epigraph is by A. J. Liebling, a political reporter for the New Yorker, and it comes from his 1963 book, The Earl of Louisiana, about Earl Long, then governor of Louisiana. The epigraph briefly discusses the Louisiana accent (a.k.a. Yat, which we talk about over in the "Setting" section) and then argues that New Orleans, as a port city, is connected to all the other port cities of the world, including those of ancient Greece.
It's not clear whether Toole is quoting this in approval, or whether it's a tongue-in-cheek claim. Maybe both; perhaps the point is that ancient Greece, often thought of as especially cultured and intellectual, may have been like Toole's New Orleans—a confederacy of scammers, hustlers, and bounders, all crawling over and bumping against each other on the harbor shore.