Jem's interest in football shows that he is now seriously moving into the adult, or at least adolescent world, leaving his former childhood games behind to take up more senior pursuits. It is noteworthy that the first time he tells Scout about his decision to try out for the football team coincides with him 'shyly' revealing his first growth of chest hair to her. Admittedly, it is scarcely visible as yet, but Scout does notice other physical changes in him.
His eyebrows were becoming heavier, and I noticed a new slimness about his body. He was growing taller. (chapter 23)
Jem is fast growing up, although Scout, being several years younger is not quite able to comprehend this yet.
As well as the physical changes, Jem is developing a sounder understanding of the world. By this stage of the novel Tom Robinson has been convicted, which has brought home the realization to him that justice and fair play do not always prevail in the adult society. But he is also learning more about the different types of people in the world, the various divisions and prejudices that exist even among people of the same race, like themselves, the Cunninghams and the Ewells, and why they all behave as they do. He is also beginning to understand the seemingly aberrant behaviour of people who live outside society, like Boo Radley. Far from regarding him as a monster, as before, he now accepts that Boo Radley may actually want to be a recluse, to never leave the house as he can't cope with the world at large.
In short, Jem is beginning to understand other people's perspectives, a sure sign of his coming to maturity. He is, in fact, now putting into practice his father's maxim that 'you never really know a man until you stand in his shoes and walk around in them' (chapter 31)