Atticus knows Scout has had a trying first day back at school, to say the least. She had a run-in with an inexperienced teacher and ended up getting a rap across the hand with a ruler for her trouble. So he tries to cheer up his daughter by regaling her with a funny story from the newspapers about some strange man who sat on top of a flagpole for some bizarre reason. This keeps Scout and Jem in fits of laughter.

But there's also something slightly subversive about what Atticus is doing. He finds out from Scout that her teacher Miss Caroline wasn't best pleased to discover that Atticus has been teaching his daughter how to read; as far as she's concerned, that's her job. So reading Scout a funny story from the newspaper is a kind of compromise; Atticus is not reading with Scout, but reading to her (and not from the kind of book that Miss Caroline would endorse, but from a local newspaper).

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.