The destruction of books leads to happiness in two major ways.
First, by destroying the books, you end the sort of inequality of intellect that you used to have. You stop having the smart people that everyone hates and the mass of "dumb" people who don't care about or understand the books. With no books, you don't have this elite who understands them.
Second, you stop having people who are insulted by the books. Beatty mentions black people and Mormons, for example. If you do not have books, you do not have anything that will say things that annoy these various groups. It is sort of like the idea of political correctness today -- you try not to annoy anyone.
The fundamental argument that Beatty is proposing is the basic idea that "ignorance is bliss." Beatty's argument is that the learning that comes out of books and the sense of understanding that emerges from them helps to create a sense of anxiety and thought about the world. Montag might be an example of this, himself. While burning books and not invested in literature, Montag is relatively happy in that he does not question anything about himself and his world. Only when he starts thinking about his world and his own sense of identity, a process that is enhanced by his own coveting of books, does Montag become more alienated from his world and thus, more unhappy. The majority of people in Bradbury's social setting are happy because of their lack of exposure to ideas, brought out through books and learning, that could cause a sense of questioning of self and of one's world. This lack of exposure helps to create a state of greater equality where all people are believing in the same notion of happiness and contentment through a lack of knowing better. Beatty's argument fundamentally underscores this point, and helps to justify his own beliefs of self.
Eliminating books also eliminates the controversy that books can generate. Beatty argues that there are too many minorities - minorities based on religion, ethnicity, political beliefs, etc. If a book offends any one of these minorities, a great controversy will be raised around it, with the minority demanding that the offensive comment be erased. In that way books create conflict. Without books, such arguments between different groups of society would be eliminated. Also, books can cause unhappy emotions. For example, one of Mildred's friends cried when she heard Montag reciting a melancholy poem. Beatty argues that such sadness is pointless, that it's stupid to get depressed for no reason like that. Finally, books generate controversy because they contradict themselves. Books contain contradictory arguments, generating more debate and conflict between the people who read them. Thus, Beatty argues, the world is better off without books.