Well, reading the article, this guy had an illegal business finding illegal music on the internet for people to play for free they didn't have rights to. A label sued him. The end.

Apple doesn't have much to do with it. They had a legal music business and negotiated to pay the labels, as they should, since the labels own the music rights. They and the labels correctly figured out their competitors were illegal, and they sued.

Should you not be able to sue someone doing something illegal just because they are in the same business as you?

For one, there's no actual evidence that actually happened.

Because in America its only anti-competitive if you can't pay off the people who decide if it is.