one who pushes the hardest for getting Boo to come out of his house. Dill understands loneliness, so that is probably why. He appreciates Boo’s situation and does not consider him a monster.

“All right then. What’d you write him?”

Dill said, “We’re askin‘ him real politely to come out sometimes, and tell us what he does in there—we said we wouldn’t hurt him and we’d buy him an ice cream” (Chapter 5).

Boo Radley really responds to these small acts of friendship. He does not mind the children reenacting his story. He seems to find them amusing and appreciates the attention. This is why he begins leaving the Finch children gifts in the tree. In Boo Radley’s lonely life, Scout and Jem are a breath of fresh air.

Summer, and he watched his children’s heart break. Autumn again, and Boo’s children needed him.

Atticus was right. One time he said you never really know a man until you stand in his shoes and walk around in them. Just standing on the Radley porch was enough (Chapter 31).

Boo Radley is childlike in his own way. He relates to the Finch children because they are kind to him. They reach out to him, so he reaches back. They give him the courage to come out of his house, which he had not done since he was a teenager. He gives them the gift of their lives when he saves them from Bob Ewell, and they give him the gift of friendship.

Jem is affected in several ways by Arthur "Boo" Radley. 

With the arrival of Dill Harris, who spends the summer at the home of his aunt's, Jem's interest in the strange recluse increases because of Dill's curiosity.

The 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.
Jem thought about it for three days. I suppose he loved honor more than his head, for Dill wore him down.

Jem runs up to the Radley house and slaps it with his palm and races back in fear. When they get on the Finch porch, they look down the street, but nothing has changed. During the summer, the children playact various scenes based on the rumors about the Radleys. After some time, however, Jem and the others become deceptive about roleplaying about the Radley family because Atticus has told them to leave Arthur alone. One day Atticus comes home unexpectedly and catches Jem, Dill, and Scout when they have been told to end such play.  Although Jem tries to deceive Atticus, Atticus uses one of his legal tricks and leads Jem into admitting that they are engaged in reenacting Boo's history. 

One night after being scolded, however, Dill and Jem decide to peep into one of the windows at the Radley house. Someone from inside the house steps out onto the porch and fires a rifle. The children scatter, but Jem catches his pants on the barbed wire. He is forced to remove them so he can escape. Having heard the report of the rifle, the neighbors come outside. When Atticus sees that Jem is not wearing pants, he asks Jem what he has done with his pants. Dill quickly says that he won Jem's pants from him when they were playing strip-poker. "Were you all playing cards?" Atticus asks pointedly. Jem intervenes and says that they were using matches. That night as he and Scout lie on the porch, Jem says that he is going to retrieve his pants from the Radley yard because he does not want his father to know that he has been deceitful again.

Despite their invasions of the Radleys' privacy and acts of imitation, Boo leaves gifts for the children in the knothole of one of the trees that the children pass each day on their way home. When they find two soap figures carved in their likenesses, Jem is very touched. He saves them in his trunk of mementos. Then, after Nathan Radley cements this hole to prevent his brother from giving Jem and Scout anything else, Jem is affected by the cruelty of Nathan Radley, and he sheds tears of anger and disappointment. Later, after the travesty of the trial of Tom Robinson, a disgruntled Jem remarks, "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." 

In the final chapters, Boo Radley demonstrates his love for the Finch children again when he becomes aware of the children's distress as they attempt to reach home but are attacked by the scoundrel Bob Ewell. Boo comes outside and intervenes, and in the struggle between the two men, Ewell dies. Afterwards, Boo carries Jem home because Jem's arm has been broken. A deeply moved Atticus thanks Arthur for saving his children's lives after Sheriff Tate explains what has happened. From then on, Jem bears the reminder of the loving intervention of Boo Radley since his injured arm is shorter than the other. 

1. Entertainment--Boo obviously watched the children as they played outside during the first summer of the book. As an isolated, childlike individual, Boo benefited from the exuberance and humor of the children's antics.

2. Human contact--When the children try to write Boo or "get him to come out," Boo seemingly takes it as someone reaching out to him, which is something he desperately wants. So, he begins leaving items in the knothole of the tree for Jem and Scout. Just the children's simple manner of trying to talk to him or write him represents the human contact that Boo has severely lacked.

3. Thoughtfulness--Scout does not realize at the end of the novel that her thoughtful act of kindness in letting Boo "escort" her back to his home is the best "gift" she could give him. She takes him back to a place where he feels safe and makes it appear to anyone who might be watching that Boo is a gentleman walking a young lady down the street.

4. An opportunity to feel self worth--In his isolation, Boo never has an opportunity under his brother's strict dominance to demonstrate that he is useful.  When he instinctively jumps in and saves Jem and Scout from Bob Ewell, he performs an action that for years later must have provided him with a sense of responsibility and worth.

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.

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.

I think you are asking how Boo Radley's character contributes to Scout's coming of age in To Kill a Mockingbird. The novel, which Scout narrates, begins when she is almost six years old and ends when she is almost nine; thus she naturally matures as the story unfolds. In addition to the age factor, Scout's maturity is impacted by the events she witnesses in Maycomb, including Tom Robinson's trial, the behavior of the lynch mob at the jail, and all of the children's dealings with Boo Radley.

Although she is frightened by Boo (and by the mere idea of him) at the beginning of the story, Scout becomes more accepting of him after receiving the gifts he leaves in the tree, which leads to a sort of friendship between Boo and the Finch children. Boo ultimately saves Scout and Jem from Bob Ewell's attack later in the novel, putting their theoretical friendship into action. The deepening relationship between Scout and Boo contributes to her maturity by encouraging her to cultivate empathy, compassion, and courage in an unjust world.

Since you are only allowed to ask one question at a time I edited yours down to just this one.

Boo Radley is responsible for saving Jem and Scout from Bob Ewell at the end of the novel.  He had every intention of hurting or even killing the children in a crazy act of revenge against the way he felt he had been treated by Atticus at the trial of Tom Robinson.

As the final chapters of the novel play out, Atticus learns that Bob Ewell was found stabbed to death and he assumes that Jem must have gotten a hold of Bob's knife and struck him with it, but Heck Tate makes it clear that it was Boo Radley who came upon the attack, and stabbed Bob in order to save the kids.  Bob Ewell had a sharp knife, but Bob was killed by a kitchen knife.  Boo's actions that night saved the children and saved Jem from having to take an action against Bob that he may never have been able to get over.  If he had in fact hurt Bob in order to save himself and Scout, he would have had to live with the guilt and the knowledge for the rest of his life.  As Scout realizes at the end, Boo has been their friend for a long time -- mending lost pants, leaving treasures in the tree, and saving their lives.  This final act allows Jem to continue to have his life and his childhood.