If we're talking about the game to get Boo Radley to come out of his house, it primarily shows that children are curious and have wild imaginations. It also shows (actually, Atticus shows them) that children are naive and sometimes unknowingly cruel. When they think of Boo, they see a monster. It is only as they get older and get instructions from Atticus that they realize he is a human being, with feelings, who happened to have had a rough life.
If we're talking about the game where Jem plays Boo, while Scout and Dill play Mrs. and Mr. Radley, the reasons are essentially the same. They are curious about this mysterious family and are trying to role play it. At this point (Chapter 4), the children still consider Boo to be a monster or a fictional character. They don't know the truth about Boo, so they make it up based on the hyperbolic things they've heard as well as the gossip from Stephanie Crawford. They have yet to consider his feelings - this is one of Atticus' most enduring lessons in the book. They're still just playing, primarily out of curiosity.
After Atticus warns them not to play the game, Scout gives her reasons why she was tiring of the game:
Atticus’s arrival was the second reason I wanted to quit the game. The first reason happened the day I rolled into the Radley front yard. Through all the head-shaking, quelling of nausea and Jem-yelling, I had heard another sound, so low I could not have heard it from the sidewalk. Someone inside the house was laughing.