Fan clubs increase the number of spectators in your stadium. This also means that you can raise ticket prices and have the same number of visitors.
You get fan clubs by solving certain challenges.
To start those challenges, you need to have Fan Mail and activate it in your inventory when you have no challenge.
You can collect Fan Mail on fridays by tapping your office if you are playing in a league (the league needs to be unfinished, so visit your office before the last match day).
The fan mail vanishes after one day. Fan club challenges have short limited run time.
The correct timing is not easy, but on the other hand fan clubs are not vital, so it's okay to need a couple of months to get them.
The number of fan clubs you can have is currently limited to 10.