Trump didn't fire Kimmel nor was he even fired. ABC suspended his show "indefinitely" (which turned out to be a few days). This was primarily because of pressure from ABC affiliates because their local markets were not happy with Kimmel, not because Trump said so, but because of pressure from customers.

I don't watch his show so I can only speak based on what I've heard but my understanding is that it was the outright lies he was telling that were controversial. For example, he was claiming that the person who killed Kirk was a MAGA supporter which was already known not to be true at the time.

The FCC is the regulatory body in the United States that can censor what goes over the airwaves. Traditionally, many of these regulations have only affected entitles that license certain parts of the RF spectrum to broadcast (ABC is one of these), not cable or streaming only entities. Normally, this would include restricting things like pornography and inappropriate language. Whether or not telling objective lies is something they can penalize you for, I don't know. However, if not, then there would be nothing really the FCC could do or that a court wouldn't overturn. Trump can say he's going to turn into a pumpkin but that doesn't mean it's going to happen. However, if he wants to speak out against ABC/Disney or whoever, he has the same 2nd amendment free speech rights as everyone else.

The only censorship that happened here was self censorship because of the negative feedback. Some ABC affiliates are still pre-empting Kimmel's show and showing other things instead. When half the country is angry about your choice of employees, that's probably going to affect your bottom line. The success of Kimmel's show ultimately depends on advertisers which depends on viewers. If there were any legitimate government censorship here, the show would not have come back after a couple of days.