Someone please explain to me why the above question is correct as written... instead of 'How do you dare?'.

It's a remnant of an old grammatical construct. It's (in that usage) called a "modal verb". Formally it modifies a phrase that includes another verb, however sometimes the entire phrase is sometimes implied: How dare you!

We get the same syntax in a lot of other somewhat-canned phrases with modal verbs:

Must you?
Need I continue?
Wilt thou take this woman... (a really long sentence followed by a short sentence, a long sentence, and a life sentence)