A breaking strategy (also known as gaming the system) is one that has been found to be objectively superior to any other strategy, often due to some flaw in the design or balance of the game, and typically resulting in one side having an insurmountable advantage once executed. Since they work better than any other strategy, there is no reason not to use them every time, and so gameplay becomes dull for both sides.
The classic example of a breaking strategy is Follow the Cop, where a known investigator and a hidden protector combine to produce a string of results. If this happens the scum must either counterclaim the investigator or kill the protector, the former guaranteeing loss of a member, and the latter being difficult and impractical. These two strategies then become the only viable strategies if the setup is known or correctly expected, the metagame suffers, and town wins the majority of the time.
For another example, a Mass Roleclaim in some Theme Games (specifically, ones where almost all of the obviously good named characters are in the game) will force the evil players to either claim names that are much more obscure, or claim a major role name and hope they manage to get the other player lynched. Some mods combat this by providing a safe fakeclaim.