Jem and Scout are Atticus's children. The story is told from the point of view of Scout, who is the younger, more impulsive and less perceptive of the two children.
A Child's Perspective
For things pertaining mostly to the children, Scout is the best possible source of information. For example, what is it like to be a kindergartner in Maycomb? What is Atticus like as a father? What is Calpurnia like as a caregiver? What are haints? For these things, Scout's perspective as a child is unique, refreshing, and often unintentionally funny.
But the two main threads of the story are Boo Radley's relationship with the children, and the trial of Tom Robinson.
Boo Radley
For the Boo Radley plotline, it makes perfect sense that we should know only as much as Scout knows at a given time. Boo starts out as a creepy, legendary figure of rumor, he turns into a mystery, then he fades into the background as more urgent things take the stage, then finally he emerges as a surprise savior. This mystery element to Boo's story could only happen if his story were told from Scout's point of view.
The Trial
For the trial of Tom Robinson, the limits of Scout's perspective make the story trickier to tell, but also easier to hear, and sometimes more poignant.
It is trickier to tell because Scout does not know the full story of what is going on. For example, there is much in the trial that she doesn't follow (though she reads the emotional temperature right). She does not really know what rape is. She does not fully realize the threat that hangs over Tom and his family.
Because of her childish ignorance, Scout is not as fully impacted by the trial as Jem is. Her little misunderstandings make a very hard story easier to bear for the reader because they give us a little distance from it.
Jem, on the other hand, is devastated by the outcome of the trial. He is old enough to follow what is going on, but as an innocent child, this is his first experience with blatant injustice. Through Jem's reaction, Harper Lee is showing us both the awfulness of racism and its commonness in Maycomb. Jem responds to the awfulness because he hasn't yet acquired an adult's cynical filter. The fact that only Jem, and not Atticus, is crying, show us how much racism is a part of everyday life in Maycomb. Atticus and Miss Maudie, though sad, aren't shocked and devastated by the trial's outcome, because they have had their whole lives to realize that this is the present reality of the world they live in.
Because Jem is not the narrator, we as readers observe his emotions at second hand and can enter into them, or not, to the degree that we choose. We are not forced to go through Jem's experience, because Scout is narrating.
One example of Scout's ignorance being poignant is the scene at the jail when Atticus is guarding Tom and a silent mob of men show up to challenge Atticus. Atticus's life is on the line, and the scene becomes even more tense when Jem, Scout, and Dill barge into this silent circle. Now the stakes of violence are even higher. Scout, however, does not realize this. She sees someone in the mob she knows, Mr. Cunningham, father of her friend Walter. Trying to be polite, she starts to engage him in conversation. As she chatters, everyone stares at her quizzically, but she does not know why. Eventually her innocent goodwill dissolves the mob, and they go home.
Scout has accidentally reminded Mr. Cunningham, and everyone else with him, that Atticus is not their enemy. He is a man they know, and a man who has helped them. But, when racial violence was on the line, this was not something they could remind themselves of without help.
Racism in Maycomb
The fact that Scout and Jem are children also gives Harper Lee latitude to let them talk and speculate openly about race, class, and family. Adult characters would not talk as openly about these things, but the kids are still trying to figure them out.
For example, in Chapter 23, Jem has a conversation with Atticus about how the law could possibly be changed to prevent the kind of injustice that Tom suffered. (Jem considers making rape not a capital crime, or doing away with jury trials in capital cases.)
Later in the chapter, Jem and Scout have a conversation in which they try to figure out what Background (being from an "old family"), which is so important to their Aunt Alexandra, really means. They cannot find a firm standard by which to differentiate their own family, from white families who are considered lower-class, or from black families or families of other races overseas. Scout concludes, "I think there's just one kind of folks. Folks." This troubles Jem, who answers, "That's what I thought, too, when I was your age. If there's just one kind of folks ... why do they go out of their way to despise each other?"
This conversation is a rich commentary on class and race. It shows how confusing the self-contradictory social rules can be to children, especially when they have also been given Atticus's common-sense remarks such as, "Every family is as old as every other family." It also shows that, despite being nonsensical, such unwritten rules are a social reality that is very slow to change. As the children get older and become more aware, they are forced to wrestle with social prejudices because the prejudices persist in being a feature of their world.