It's a really rude way for someone to try to block the other person from continueing to talk or to stop the whole conversation.The person isn't giving a single reason why they want the other one to shut up, because they're probably annoyed or offended.

I don't think I've ever heard anyone say exactly that, though I can certainly imagine it being said.

If one were told to do anything, and one asked why before complying, the impatient, authoritarian response could be: Because _______, that's why.

The responder is saying that he doesn't have to give any explanation or reason for giving the order.

Sit down.

Why?

Because "sit down", that's why.

In the English of SE Pennsylvania where I live, I think I would more likely hear:

"Because I said so", but the "Do it, and don't question why" meaning is the same.