Fahrenheit 451 is a book about book burning. It is also a book about the knowledge that can be gained from reading books.
A central conflict in the book is between Montag and Beatty. Montag is curious about books. He's on the fence about whether to burn them or not. He's a smart guy and knows that if he doesn't have a clear reason TO burn them, maybe he shouldn't. Beatty, on the other hand, firmly believes that books need to be destroyed. His reasoning can be convoluted at times, like his arguments, but Beatty basically believes that books need to be destroyed because they don't offer people anything useful and constructive. In Beatty's mind, books confuse people more. They bring up more questions than answers.
It's fitting that people brag about not reading the book, because they are doing exactly what Beatty supports. Beatty says that reading encourages people to think for themselves, and that is why books are dangerous. A society without books leads to an easy-to-control population of people. People that don't read the book are basically bragging about the fact that they are ignorant, non-thinking followers. They aren't independent. They are dependent on what other people tell them about and tell them to do. It's ironic, because they think that they are being independent thinkers, when in reality they are further deepening their ignorance.