It would take a couple of hours to adequately answer your question, but it's okay, the information is all already available here. It's all free but will take you some time to read what's on the forum.
Mystery shoppers run their own small businesses working for themselves. They perform assignments for various mystery shopping companies as Independent Contractors. It requires a large expenditure of time initially to research how it works and to get started. If you go to the New Mystery Shopper area, there is a lot of information available on how to do it. A list of legitimate companies is at the bottom of the page, the blue link, Official List of Mystery Shopping Companies. Each shopper decides how many companies to register with and which ones. Each company has a registration process the shopper can follow by going to their website.