The short version is that, as of a change a few seasons ago, you get paid half your sponsorship income at the beginning of the week. The other half depends on your match ratings in each game. If you meet the "minimum" match rating requirement for your division (or - more rarely - if you rate below the minimum but still win the game, you get the other half of your sponsorship. If you lose with a match rating below the standard, you don't get the money. The intention is to prevent people from "tanking" or getting a massive financial windfall from selling up while in a high league.
The full details are on the finances page in the game manual, as Boscorp said. The rating requirements are very lenient, honestly; you're not going to fall below them by accident or promote into a league where you can't rate high enough.
