The 'big bad john story' is often about a character named John who is perhaps seen as tough or menacing in some way. It could be a story of his adventures, misdeeds, or how he impacts the people around him.
Well, the 'big bad john story' might be a traditional or local tale. Maybe John is a bully in the story, causing trouble in a small town or a community. He could be interfering with the normal lives of the people there, like scaring the local merchants or being rude to the townsfolk. And then the story could be about how the people in the town deal with this 'big bad john', whether they fight back or find a way to change him.