“I rest my case” means I have finished my argument. You can use it in exactly that circumstance - after you have completed making your argument. You could also use it after your opponent concedes a point in an argument. “I rest my case” (“That point is so important that there is no point in arguing any more. I should not need to argue any more. You should agree with me now) It’s used when you are very confident in your argument.

When I want to say that I am giving up on something

It really helps my understanding.Thank you so much.

"I rest my case" is used to imply that a point you are trying to make has just been shown as true. It's used when you make a statement about somebody or something and then that somebody/something (usually through an action) demonstrates what you said is true or accurate, and then you say "I rest my case!". So imagine I say to you "You're always looking at your phone when I talk to you" and you say "What are you talking about?! I listen to you....oh, hang on, I'm getting a text message" and then you look at your phone. I'd then say "I rest my case!", because your action just proved that I am right. Or imagining that I was arguing that somebody is always doing something and you don't believe me, but then that other person walks into the room and does exactly what I was talking about, I could say "I rest my case!". It kind of means, "See! I told you so!" in this sort of situation. Or, "See! There is evidence right there that I am right!".