In "To Kill a Mockingbird", how do Scout and Jem contribute in Boo Radly eventually coming out?
It is hinted, at the end, that Boo was an indirect participant in much of their childhood. He probably observed them, and all of their adventures and exploits, from the safety of his window; as a result, probably grew very fond of them. He even extended a sort of hand of friendship when he started putting things in the old tree for them. He obviously cared about them enough to, when he sensed them in danger, come out to help them. It wasn't the first time; during the Miss Maudie's fire, Scout discovered that a blanket had been placed around her shoulders to keep her warm. Atticus hinted that it was Boo that placed it there.
But the second time he emerged, it is in reaction to Jem's scream; he hears it and is prompted to more aggressive action. He helps get them home, and sits during the resulting decisions and chaos. And they all react as if it is completely normal, as if he is an old family friend, which, in a way, he was.
The direct answer is, when the two are endangered by Ewell, Boo takes action to assist them in their peril. He hears the cries of Jem and the sounds of conflict, and is prompted to action by those noises.
Prior to this scene, however, we have been led to infer that Boo is responsible for placing trinkets, coins, and "treasures" in his tree's knot hole for the children, and that he also blanketed Scout at the scene of Miss Maudie Atkinson's house fire. Atticus says that Boo is responsible for the blanket, even though Scout never saw him at the scene itself.
The main way the children are responsible for Boo's revelation to the neighborhood is through their own endangerment, however.