All clubs start off with a small number of hard core supporters. Your results on the field will affect the rate at which your supporter base increases.
The number of people who attend your matches is heavily influenced both by the number of supporters in your club and their current mood. Winning competitive matches (ie the results of friendlies are ignored) will improve your supporters' mood, causing more supporters to join your club each week. Conversely, losing competitive matches will cause the number of supporters joining each week to reduce to a trickle. Lose enough matches and supporters might even start to leave your club!
Once the number of supporters in your club reach a certain number, simply winning matches won't be enough to cause your supporter numbers to continue increasing. After a certain point, promoting into a higher division will then be the only way to cause your supporter numbers to continue to grow.
Supporter mood is reset to jovial at the end of each season.