Because if there wasn't a fine/permit, you would have all the public tennis courts camped at at all hours of the day by teachers giving paid lessons to students.

Tragedy of the commons.

City parks departments don't want random instructors to book courts to teach, they want them to go through their own instructor staff, and want the courts to otherwise be used for people playing the game. I don't think it's an uncommon practice, I think the NYC Parks department works the same way.