This line occurs in Chapter 12. Cal has brought Jem and Scout to her church. The congregation is taking up a collection for Helen, Tom's wife, because she is having trouble getting work while Tom is in jail. After the collection is taken, Reverend Sykes says that they do not have enough money. He actually orders someone in the back of the church to close and lock the doors until more money is given. When Scout says Reverend Sykes intends to "sweat the amount due out of his flock" she means that Sykes is literally going to make them sweat until he has the full ten dollars. (The church is hot.) But she also means this metaphorically. Reverend Sykes will wait until the congregation gives more money. Here, "sweat it out" is a metaphor for waiting during a difficult situation.
Scout also uses the word "flock" here. This is a figure of speech for the congregation (people that regularly attend the church). This usage of "flock" dates back to at least the Old Testament. The shepherd (Reverend Sykes in this case) guides his flock in religious and spiritual matters. In this instance, he is trying to "sweat" (wait) the money out of his flock.