To give oneself mighty airs: to assume, in manner, appearance, and tone, a superiority to which you have no claim. The same as Air, manner (q.v.).
The plural is essential in this case to take it out of the category of mere eccentricity, or to distinguish it from “air” in the sense of deportment, as “he had a fine, manly air,” “in air was that of a gentleman.” Air, in the singular, being generally complimentary, but “airs” in the plural always conveying censure. In Italian, we find the phrase, Si da dell árie.