Jem asks Scout to not antagonize their Aunt Alexandra because if she becomes upset it places stress upon their father, who is already under great social pressure with the forthcoming Tom Robinson case.
In Chapter 14 Atticus is taken from his perusal of the newspaper by Scout's abrupt question about the meaning of the word rape. She explains to her father that she was told by Calpurnia to ask her father what the word means. Her mention of Calpurnia brings out the fact that she and Jem attended church with Calpurnia while he was in Montgomery. Further, Scout wants to know if she may go to Calpurnia's house some afternoon.
When Aunt Alexandra hears that Scout wishes to go into the black neighborhood with all that is going on in Maycomb, she adamantly says to Scout, "You may not." Her peremptory tone causes Scout to retort, "I didn't ask you!"
When he hears Scout being disrespectful, Atticus demands that Scout apologize to her aunt. Further, he instructs Scout to obey her aunt as long as she is in their house, just as she should obey Calpurnia. Embarrassed, Scout leaves the room only to hear a discussion between her aunt and her father on both Scout and Calpurnia.
Because this discussion ensues and because Scout has bothered her father when he already has so much on his mind, Jem asks his sister, "Scout, try not to antagonize Aunty, hear?"
Jem is growing up, and even though he isn't fond of Aunt Alexandria he does not want to see drama. Also, Aunt Alexandria often lectures Atticus about how Scout should act. Jem is probably trying to protect his father.
Jem is gradually comprehending maturity as he steps into adolescence. He is gaining the emotional capacity to appreciate the domestic compliance Aunt Alexandra impels into the family and instead of being unappreciative, Jem starts to look into Auntie's tendencies to do good to the family.