Here, "business" doesn't mean "money-making enterprise." It literally means "busy-ness." It means anything important that someone is doing.
"My business" means things that I'm in charge of, things I manage, my duties, my responsibilities. "Your business" means your duties.
"Mind your own business" is an indirect way of saying "stay out of my business." It carries the implication that not only is someone trying to interfere in your business, but that while he is doing it, he is wasting time that he should spend on his own duties and responsibilities.
Suppose you are playing chess. Next to you, two other people are playing chess. One of them gets up, walks over, looks at your board and says "You should move the knight here." You say "mind your own business." It's not his game to play, it's yours. He shouldn't be over at your board telling you what to do. He should be back at his own board, paying attention to his own game. He should be minding his own game--his own "business."