In To Kill a Mockingbird how does the school setting give the reader an important exposition about southern culture?
The school is a great place for the entire spectrum of Southern culture to be revealed, because children from all walks of life, from every demographic and socio-economic category all go to the school. So, each type of person and every quirk of the culture can be represented through the children in one small classroom. Even better is if the teacher is new to the area, and so must be "initiated" into the culture, either through direct instruction (as Scout attempts to do), or through personal experience (as happens when Miss Caroline tries to instruct Burris on his hygeine).
In the classroom, we get to see the rather infamous Ewell children first-hand, along with unsavory details about their appearance and dress. We learn that they only attend the first day of school, so as to avoid the authorities and their meddling, and that education is of little value to the entire family; we also learn that their father is a no-good drunk. We also learn all about Chuck Little, who was also poor, but "was a born gentleman." Then there is Walter Cunningham, also poor, but with a proud father who refuses governmental help. All of these different categories of people have their own way of living in the south. Scout herself tries to inform the naive and uninformed Miss Caroline about the intricate ins and outs of Maycomb culture, but she doesn't receive it too well.
The classroom provides first-hand witness to some of the different social classes that exist in the town, and it does so in an amusing and harmless way, as a sort of precursor to the more intense and serious action that is to happen later on in the book. Later, we see these same families behaving in rather unfortunate ways, and it helps to have a bit of background, through the school, before those events occur. I hope that helps; good luck!