The kind of happiness that Beatty is espousing (in Ray Bradbury’s “Fahrenheit 451”), when he says that "We're the happiness boys, the Dixie Duo" is an unfruitful pseudo-happiness that seeks to limit a person’s thinking. In essence, Beatty is saying that the firemen have a responsibility to keep books out of citizens’ hands so they cannot learn about alternative views and ways and therefore go against the dictates of the totalitarian state. Totalitarianism and its suppressive policies are the environment of this classic novel.
The type of happiness espoused here is a “don’t rock the boat” happiness. It seeks to keep the citizenry at bay so they do not engage in anti-government thinking and actions – revolt. The government and its agencies, which include the fire departments, want to tell people how to think. They want the citizenry to believe that all is well. They know that if they allow people to explore books and the imaginative thinking of others that it could spell their (the government’s) doom. They know people will realize that they’ve been duped all along; that the society in which they live is not a Utopian society as promoted by the government.
This false happiness is accepted by the majority of the populace because they know no better. They have not investigated books because they have made the conscious decision not to do so; but a very small minority have made the decision to track down and investigate books.
The majority of the citizenry is somewhat happy in their complacency because they know that if they tow the ‘party line’ they will not be punished in any way. They can continue to live their mundane lives – without engaging in dissenting thought – and their decision to avoid creative thinking (that challenges government policies) will permit them to live a peaceful life free of any punishment from the government.
As a result, it is a perverted kind of happiness that they ‘enjoy’ and it’s a happiness that does not bear good fruit in their lives. They, if they follow the government’s dictates, cannot promote policies and laws that they believe are better than what the government is promoting and making law. Fundamentally, they are held hostage by this false happiness as they are not allowed to deviate from what the government wants them to think.
The "happiness" that Capt. Beatty is referring to is the bland happiness that comes from the lack of controversy and the lack of having to think. In this scene from the book, he is telling Montag why their society is the way it is; how people wanted everything to be politically correct and free from insult to anyone and how people wanted everything to be faster and easier. He says that he and Montag (and all firemen), because they enforce the anti-book law, are the ones who bring happiness to people because they help to make sure that no one reads anything upsetting or that takes time away from other pleasures.