Jem and Scout turn Boo's life into a game for a few reasons. First of all, they don't really understand Boo (Arthur) or fully appreciate what happened in his life, how he may have been mentally abused by his father and, because of those things, the reasons why he is forced and/or chooses to remain in the house all of the time.
They are children and subject to the whims of their imaginations. Atticus tells them to leave Boo alone and provides the sober, objective facts of Boo's life but the children are drawn to the more outrageous stories about Boo (some coming from the gossip, Stephanie Crawford) because it appeals to their wild imaginations. Scout and Jem are also encouraged by Dill's curiosities. Dill certainly is intrigued by the mystery surrounding Boo and he is often the one instigating the games they play. In the very first chapter, Dill gets the idea to make Boo come outside. Dill bets Jem that he won't be able to get farther than the Radley gate. Eventually, Jem takes the bet.
Our first raid came to pass only because Dill bet Jem The Gray Ghost against two Tom Swifts that Jem wouldn’t get any farther than the Radley gate. In all his life, Jem had never declined a dare.
As the novel continues, Jem and Scout (and sometimes Dill) continue to play games involving the Radleys because they are curious and because there is so much mystery about why a grown man would stay inside his house.
After experiencing the faults of adults and the difficulties brought on by the trial, Jem has a realization. At the end of Chapter 23, he says to Scout:
Scout, I think I’m beginning to understand something. I think I’m beginning to understand why Boo Radley’s stayed shut up in the house all this time… it’s because he wants to stay inside.