“Because I said so, that's why.”
You've asked your son to clean his room. He asks you why he has to do that. You're annoyed with him, so you don't feel patient enough to explain why. You just say this.
Because I said so, that's why.
When parents tell their children what to do, the children don't always obey. Sometimes they ask their parents why they have to do something:
A: Go to your room!
Sometimes parents get annoyed that their child isn't obeying them, so they answer with this phrase:
Because I said so!
This means that the child is supposed to obey and not ask questions.
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?