They are funded by private groups or companies. You are right, it does not appeal to enough fans. I think the average person does not enjoy problem solving puzzle games like chess. And if they do like it, very few are enthusiasts because of it's complexity/mental toll.
Otherewise, I think Fide or the USCF would have to start a major campagin over the course of decades , similar to the NBA, to garner a large fan base for chess. It would also help if Magnus Carlsen, being the #1 player in the world, could get better sponsorships and advertisements/commercials.
How are chess prize-pools funded (the government, corporations, etc.??) and why does chess not have the million or hundred thousand dollar salaries of other sports?
Does it just not appeal to enough fans/viewers for which advertisers might fund events and prizes?