Okay kiddo, so imagine that there are some people who get paid to make sure that everyone follows the rules. They're called "attorneys." Sometimes, the government can't catch everyone who's breaking the rules, so they let regular people step in and be their own mini-attorneys. We call those people "private attorneys general."
So let's say you noticed that some grown-ups in your neighborhood were littering and it made you really mad. You might be able to take pictures of them littering and send the pictures to the government. If they don't do anything about it, you could actually start a lawsuit against those grown-ups that litter. As a private attorney general, you're basically trying to help the government enforce the rules.
Sometimes, people become private attorneys general for really important things, like fighting discrimination or protecting the environment. They do this because they believe it's really important to enforce the rules and make sure everybody is being fair and safe.