There are two levels to the sentence you highlight, the denotation or literal meaning and the connotation or poetic meaning. On a literal level, this sentence is just a way to say that the boy is very dirty and that he looks intently at his teacher. On a poetic level, this sentence contains figurative language that elaborates the themes of the whole novel.
First of all, consider the literal level. Why use this strange manner of speaking? What is a 'fist-sized clean space' in someone's face? Bear in mind that one's eyes are never dirty. Even a coal-miner with black dust all over his face will have white eyes; the eyes continuously clean themselves. I imagine that Burris also has the folds of skin around his eye-sockets and eyelids showing lighter than the rest of his grime. So much for 'clean space'. Now, to peer is to look intently, and a fist-sized clean space is an area free from dirt that is roughly the size of a fist. A strange way of putting it, but it makes sense on the poetic level.
This sentence describes the actions and appearance of Burris Ewell while also adding detail to his character and his relationship to the teacher. Burris is peering at the teacher because he is scrutinizing her; this is a confrontation, and one faces the enemy, stares them down, before a battle. Miss Caroline has just challenged him by instructing him in the proper way to treat lice. He stands up and rebels, rudely laughing at her and disdaining her hygiene advice. Later in the book, he will insult Miss Caroline by calling her a snot-nosed slut.
Keep in mind that Bob Ewell, Burris' father, is the main villain of the book: a racist, ignorant 'white-trash' patriarch. Perhaps now it makes sense that we read of a 'fist-sized' clean space and not, say, an 'apple-sized' space. Burris Ewell, like his whole grimy family, exudes hatred, hence the mention of a 'fist' (a kind of weapon). Though they may be so-called 'white' people (who have skin colors ranging from pink to light brown, if we speak scientifically), their souls show more darkness than light.
Recall the context of this passage in chapter 3. The passage is a scene in the schoolhouse that young Scout observes without participating in. Scout and Miss Caroline learn a couple things about their classmates and the unspoken social ties of their town. As readers unfamiliar with this southern world, we learn too. It is the second scene in chapter 3 and it shares similar themes with the first, which is worth examining in order for the contrast.
The scene between the filthy boy and the teacher is an example of a low-class, ignorant person coming into conflict with a higher-class teacher. By the end of it, the Ewell boy has fled the school, spurning his elders and the principal's authority. Miss Caroline is in tears, shamed before her schoolchildren. Her efforts at education, despite her own refinement, were embarrassing failures. This is almost the polar opposite of the scene that introduces chapter three. In that scene, a poor but polite boy will unwittingly make a social blunder at the table of the refined lawyer, Atticus Finch. Atticus, the hero of the book, will graciously allow the boy to break the rules of his own table out of charity and kindness, but Scout will spoil it and make her poor guest bow his head in shame.
That scene begins the chapter. We find Scout is bullying a boy, Walter Cunningham. Scout's older brother Jem breaks up the fight and invites Walter to join them at home for lunch. Throughout this scene, the tomboyish Scout learns about social grace from her brother, father and the maid Calpurnia. In particular, Scout scoffs at her guest Walter: he pours molasses all over his plate, dousing his meal in syrup.
Here is the scene, for the record:
Atticus was expounding upon farm problems when Walter interrupted to ask if there was any molasses in the house. Atticus summoned Calpurnia, who returned bearing the syrup pitcher. She stood waiting for Walter to help himself. Walter poured syrup on his vegetables and meat with a generous hand. He would probably have poured it into his milk glass had I not asked what the sam hill he was doing. The silver saucer clattered when he replaced the pitcher, and he quickly put his hands in his lap. Then he ducked his head. Atticus shook his head at me again. "But he's gone and drowned his dinner in syrup," I protested. "He's poured it all over-"
It was then that Calpurnia requested my presence in the kitchen.
She was furious, and when she was furious Calpurnia's grammar became erratic. When in tranquility, her grammar was as good as anybody's in Maycomb. Atticus said Calpurnia had more education than most colored folks. When she squinted down at me the tiny lines around her eyes deepened. "There's some folks who don't eat like us," she whispered fiercely, "but you ain't called on to contradict 'em at the table when they don't. That boy's yo' comp'ny and if he wants to eat up the table cloth you let him, you hear?"
"He ain't company, Cal, he's just a Cunningham-"
"Hush your mouth! Don't matter who they are, anybody sets foot in this house's yo' comp'ny, and don't you let me catch you remarkin' on their ways like you was so high and mighty! Yo' folks might be better'n the Cunninghams but it don't count for nothin' the way you're disgracin' 'em- if you can't act fit to eat at the table you can just set here and eat in the kitchen!"
Scout does not realize that southern hospitality requires her to ignore the gluttonous, uncouth way that Cunningham uses the syrup. She knows she would surely be punished for behaving so, and sense injustice. What she does not realize is that she and Cunningham come from different worlds. Her father, Atticus, is the hero of the book and one of the all-time greatest moral exemplars in American literature. Atticus does Walter two kindnesses that afternoon. First, he generously shares his food with the poor boy; second, he avoids shaming him by bringing up their difference in social class. Yet this is exactly what Scout does when she asks 'what in the sam hill' (which is a southern exclamation similar to "what in the hell", but not technically a swear-word) he is doing pouring the syrup so boorishly.
This is what immediately proceeds the scene wherein we find your quote. Like Scout, Miss Caroline learns a few things about the world that morning. Though Miss Caroline is an educated adult like Atticus Finch, she is less savvy than him. She fails to avoid an embarrassing confrontation with a child of much lower social status. This pair of scenes connect to the overall themes of moral courage and the difference between a rigid-but-inhuman justice and a flexible-but-graceful justice.
I don't know why I went on so long on this point. Your question inspired me to revisit this wonderful book. In a masterpiece of literature, even a little sprout of a sentence, a single word or leaf, has deep roots connecting it to the greater themes of the novel. If we pull on it, look how far we can follow it!