This passage from Chapter 3 of To Kill a Mockingbird comes at the end of the chapter and the conclusion of a disturbing day for Scout who has some rather negative experiences with her new teacher, Miss Caroline. In contrast to Miss Caroline who, after being consoled by the children for the insults of the lice-ridden Burris Ewell, "mystified the first grade with a long narrative about "a toadfrog that lived in a hall," Atticus Finch amuses his children by reading from long columns in The Mobile Register. Here, then, is again illustrated the ineffectiveness of Miss Caroline to understand the children.
In the context of what has happened to Scout, her "fits" are not only amusement, but probably also a release of emotion from the stress of early incidents in the day. In the comfort of the nightly routine--"Jem and I were accustomed to our father's last-will-and-testament diction"--and at home near her beloved father, Scout releases her surpressed anxiety of her school day now in "fits" of laughter.