“I don't want to bother him.”
Bother means to disturb or inconvenience someone. Tina does not want to keep contacting a person who has not replied and make them uncomfortable.
A2-13 / CEFR A2 / EPISODE 13
Talking about waiting for a reply
SceneTina is worried because her messages have not been answered, and asks Frank for advice.

WORDS & CONTEXT
Bother means to disturb or inconvenience someone. Tina does not want to keep contacting a person who has not replied and make them uncomfortable.
Give someone time to ... means to allow someone time to do something, and keep it short means to make it brief. Frank first advises waiting for a reply, and suggests that if Tina still wants to write tomorrow, she should send just one short message. He is not telling her she must send one the next day.
Just in case ... means as a precaution, in case something happens. Here it explains why Tina wants to bring her phone: in case he replies. The humor is that she is already worrying about a reply right after promising to stop checking her phone tonight.
Literally "I mean it," this is a set phrase in conversation for stressing "I'm serious; I'm not joking." Tina has just promised not to check her phone, yet she tries to bring it along to wait for a reply, so Frank stops her: "Really, leave it."