This fund-raising football game is seen in Chapter 10. All we can really say for sure is that the Methodists challenged the Baptists as a way to try to raise money to pay off their mortgage. There is nothing in the text to tell us exactly how the game was supposed to help pay off the mortgage.
Generally, though, there are a couple of ways that institutions use games like this to raise funds. First, they can charge admission to the games. We know that the Finch family went to watch and probably a lot of other families did too. If they all paid admission, the Methodists could get money that way. Second, they might be able to get businesses to sponsor the game. The businesses could have advertisements up at the game or in a program or something like that.
These are ways I've seen games used for fund-raising and I assume they were doing one or both of these in this case.