Grammar of 'dare' in this example

Solution 1:

Dare (like need) is a semi-modal verb, which means that in certain environments (see Negative Polarity), of which Yes/No Questions are one, dare may be used as a modal auxiliary verb.

If you substitute any other modal for dare in the sentence, you'll see what I mean:

• The pizza was nice but, may I say it, the salad was awful.
• The pizza was nice but, can I say it, the salad was awful.
• The pizza was nice but, might I say it, the salad was awful.

These aren't all idioms like dare I say, and of course they don't mean the same, but they illustrate how modals work in questions. The only unusual thing is using dare as a modal verb here.