By Chapter 11 of Harper Lee's To Kill a Mockingbird, Scout and Jem begin fighting because racial tensions are rising in Maycomb due to their father's decision to put his all into defending Tom Robinson. As a result, Scout and Jem are beginning to suffer more and more ridicule, ridicule that is especially wearing out Jem, leading to fighting between the two children, and this fighting continues to progress in Chapter 14.
In Chapter 11, Scout notes that Jem typically has a "naturally tranquil disposition and a slow fuse," meaning that Jem usually has a very naturally calm temper, as opposed to Scout's own temper; yet, when Mrs. Dubose exclaims, "Your father's no better than the niggers and trash he works for!," Jem loses his temper, leading to the children's first major fight in the book. Jem grabs the baton he had just bought Scout and uses it to whack all the flowers off of Mrs. Dubose's camellia bushes. He then breaks the baton in two while Scout shrieks. Jem becomes even more physically violent with Scout including yanking her hair, kicking her, making her fall flat on her face, picking her up roughly and dragging her away from the scene of the crime. All of Jem's uncharacteristic violence shows that the town's racist ridicule of his father is wearing him emotionally thin.
Similar violence ensues in Chapter 14. During this chapter, Atticus and his sister Alexandra begin quarreling about racial issues. More specifically, Aunt Alexandra says she no longer wants Calpurnia working for them and looking after the children since Alexandra thinks it's improper due to Calpurnia's race. However, Atticus refuses to fire Calpurnia, declaring Calpurnia to be invaluable to the Finches and a member of the family. When Atticus and Aunt Alexandra begin quarreling, Jem warns Scout "not to antagonize Aunty," because he is concerned about giving their father more to worry about than Robinson's case and sees what a tribulation Robinson's case is due to the ridicule he suffers on his own. When Scout refuses to listen, feeling insulted that Jem would have the nerve to boss her around, Jem threatens to punish her, ensuing in a second physical fight between the two siblings.
For Jem, the cause of the fight is again due to the persecutions he is suffering as a result of racism in Maycomb. However, Scout makes the first physical attack, and her reason, in addition to the stress caused by racial tensions, is because she has been feeling distant from Jem ever since he began to grow up as a result of his experience with Mrs. Dubose. As a result of this distance, Scout feels younger and inferior to her brother. Attacking her brother and inciting him to attack back gives her the sense that, as she phrases it, "We were still equals."
Hence, all in all, Jem and Scout begin to fight because ridicule from racial tensions is wearing Jem down emotionally. In addition, Jem is growing up, which makes Scout feel inferior to him, leading to further fighting.