Don't you mean when he tells on Dill when the Finch children find him hiding under the bed?
Jem knows it will probably get Dill into temporary trouble to be "found out" (Dill was unhappy and bored at home and had run away to Maycomb), but he is old enough at this point to know that the adults need to be informed of Dill's whereabouts. Dill couldn't just camp out at their house while he was listed as missing. Also, his mother was sick with worry.
In a way, this was a distress signal from Dill that his mother and stepfather had been neglecting him ever since his mother had married. Dill, an illigitimate child (sent off to his aunt's place during the summertime and enjoying it), also needed a place in their hearts to "belong."
This is a fairly broad question, as older brothers in general tend to do a lot to make their younger siblings upset. I can give you a few examples, so I hope that helps.
1. As Jem gets older, he starts acting more superior than Scout, treating her in the same way that Atticus, her father, treats her by trying to tell her what to do.
2. Makes fun of her, calling her "Angel May" when she is hesitant about going up to the Radley's house.
3. After Jem is angered by Mrs. Dubose, not only does he destroy Mrs. Dubose's bushes, but he also breaks Scouts brand - new baton.