This idiom or expression like sentence means:

I’m finished arguing.
I’m not arguing any more.
I’ve said all I’m going to say about it. I have nothing more to add.

Also, it is sometimes used to conclude that one is right.

A. Where were you when the robbery took place?
B. I don’t remember!!
A. You don’t remember? Well, I rest my case. (As you can see, I was right!)

Also see:
Case closed.

This idiom or expression like sentence means:

Finished.
No more discussion.

When someone says “I’m not buying a new car, case closed,” they mean something like:

My decision on buying a new car is final.
I’m not buying a new car, and I don’t want to talk about this subject any more. Etc.

Also see:
I rest my case.