How you love to hate me?
This is not a subject question, but why is it correct and we don't say
How do you love to hate me?
I have same problem with this question:
Why she said that?
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The only way that
How you love to hate me!"
is grammatical is as an exclamation, not as a question. It expresses surprise at your hate for me.
Why she said that.
is not a grammatical sentence. It can be grammatical as a clause, a part of another sentence, for example:
I don't know why she said that.
"How you love to hate me!"
The above is not a question. It is an idiom used in English to provide emphasis. It is perhaps best described grammatically as an exclamation.
Here are some similar examples.
What a good person he is! (He is a very good person!)
Why it's you! (I am surprised to see you!)
How you love to hate me! (You hate me and you love the fact that you hate me!)
Your title is "Why this sentence is correct?" however that title is incorrect in English. You should ask "Why is this sentence incorrect?"
Similarly "Why she said that?" is a mistake (that learners of English often make). The correct version is "Why did she say that?"
I see that @Jack O'Flaherty answered at the same time as I did. I am glad to see that our answers are similar.