What feeling is expressed in the sentence. "How dare you talk to me like that?"

Emotive interjections

An emotive interjection is used to express an emotion or to indicate a reaction to something. For example, the emotive interjection “ew” is used to express disgust.

Curse words, also called expletives, are commonly used (in informal contexts) as emotive interjections to express frustration or anger.

Examples: 

Ew. I’m not eating that.

Yay! I’m so excited to see you.

Yum! This apple pie is delicious.