If someone asks you a "Why" question, and you're annoyed that you have to answer, you might answer like this:

A: Why don't you come home first.

B: Because I don't have time, that's why!

Adding "That's why." at the end of your explanation shows that you're annoyed and don't want to explain more.

But "That's why ___" followed by a reason doesn't express annoyance. For example, a patient mother might say this to her child:

Your room is really messy. That's why you have to clean it, OK?