What does Jem want Scout to kill as they are sleeping on the porch and why might he not want her to kill it? What does it symbolize?

Jem and Scout are about to go to sleep on their cots on the back porch. Scout sees a roly-poly bug and goes to kill it. Jem stops her, saying the bug never did anything to harm her. Scout heeds his request and carefully takes the bug outside, noting internally that if anything, Jem is becoming more like a lady than she is. As she returns to her cot, she thinks of Dill and remembers his story of the day Tom Robinson died in late August.

The roly-poly incident is yet another example of Jem's increasing maturity. Having witnessed Tom's trial and his family's reaction of his death, Jem has an even greater sense of the need to protect the innocent. Therefore, the roly-poly bug is a symbol of the weak and oppressed who are often "stomped on" by society.