These are the exact instructions that Jem gives to Scout at the beginning of chapter two:
"during school hours I was not to bother him, I was not to approach him with requests to enact a chapter of Tarzan and the Ant Men, to embarrass him with references to his private life, or tag along behind him at recess and noon. I was to stick with the first grade and he would stick with the fifth. In short, I was to leave him alone."
These instructions seem pretty typical of an older sibling who doesn't want his younger sibling to embarrass him to death at school; school is a cruel place where kids make fun of each other for any little thing. Scout, with her unusual bluntness, tactless curiosity, and close relationship to Jem-who was quite a bit older, was primed to go in there and do something that was sure to get Jem made fun of. Sure enough, at recess, she is about to lay into Walter Cunningham when Jem has to intervene. So, Jem's instructions were born from a desire to keep his dignitiy in tact at school, and a good understanding of Scout's personality.
Jem essentially told Scout that she was to leave him alone. He clearly did not want to be bothered by his little sister. She had to stick with her own first grade whilst he would stick to his fifth.
Jem's demand is an indication of how well he knew his sister. He is for example, well aware that Scout does not always think before she speaks and he dis not want to be embarrassed by any foolish remark or question she might ask. A good example of this was when, in chapter one, when Dill told them about not having a father. He was clearly embarrassed about the fact and when asked whether he was dead, he admitted that he was not. At this, Scout bluntly asked,“Then if he’s not dead you’ve got one, haven’t you?” Dill could not hide his discomfort and Jem told Scout to be quiet.
Clearly, Jem wants to avoid this kind of situation at all costs. Furthermore, it is clear that Scout does not yet know when to mention or not mention any detail that might also put him in an awkward position, such as the time when he told Dill that he wasn't afraid of the Radley house since he had walked past it many times before. To this Scout had said, "Always runnin'," which, obviously, discounted his assertion of not having any fear. Scout's presence at the school would be a difficult situation to cope with already and having her around him could make him feel uncomfortable.
This, however, does not mean that Jem is embarrassed about his sister per se, for he later, for example, intervenes during her altercation with Walter Cunningham. It is just that, as an older brother, he needs to have his own space - as he always had - and not have his position compromised.
she is not to make a fool of herself, not to embarrass herself, or her family. She is to watch her mouth...what she says