It means: getting vibes that she's interested in him; assuming she likes him; etc. 'Getting vibes' is a phrase I've heard a lot. But I've never heard 'getting airs' before, so I assume it's either a quirky expression invented by the author of what you're reading, or known only to people who read a lot of fiction and/or an old, outdated expression quite redundant today. Still, by reading the whole quote it's possible to assume this is what it means. There's enough context there to assume 'airs' means 'vibes of reciprocal attraction'.

He was told that he shouldn't become too proud/smug/elevated/self-appreciative.

Hi Richard, Excellent description! I thought I should look up the origins of this, but now I don't have to. :)

Yes, you are correct. It is the same as 'putting on airs'. The word 'air' in this case is a use of the French word air - meaning 'appearance / bearing / manner / behaviour'. The saying is used to describe when someone acts arrogant or pretentious. When they pretend to be better than other people. Some similar word/phrases you may read: Your 'place'. As in, 'remember your place' Your 'station'. As in, 'don't get ideas above your station' Where place/station refers to your social / economic status. These types of phrases are quite old fashioned now. A lot of native speakers might have heard them, but not understand what they mean. They are mostly from older times when there was a very rigid class structure.