“What did you just say to me?”

You’re yelling at your daughter to clean her room. She says, under her breath, “I hate you!’. Now you’re extremely angry, and you can’t believe she said that. You say:

What did you just say to me?

What did you say to me?

If you didn't hear what someone said, you can ask:

What was that?

What did you say?

But asking "What did you say to me?" has a completely different meaning. You say this when someone insults you, challenges you, or says something rude:

A: Shut up.

B: Excuse me? What did you just say to me?

You should say this in an angry tone of voice, and stress the word "what".

(someone) just (did something)

One of the meanings of "just" is "just now". In other words, it means something that happened right before you spoke. For example:

He just left.

I just finished.