The word "break" means something fell apart into pieces, or stopped functioning properly. To "break" an egg, for example, or "he's going to break that vase if he drops it".
But it also describes a sudden change in direction. For example..."The pilot was told to break to the right".
"To Break Bad" means that someone is changing the direction of their life, from being a normal law abiding citizen to being a criminal (a BAD person). In the case of the TV series, "Breaking Bad" refers to Walter White, the high school science teacher who becomes a drug criminal.