Basically, it means he acted sophisticated so he could impress her parents. You might say he pretended to be sophisticated when he is in fact not the least bit sophisticated, but it's possible he really is sophisticated but usually doesn't show it. There are some finer elements to it. "An air of" suggests that the performance was subtle. Think of a perfume wafting through the air. It may have an effect on you before you are even aware that you are aware of the scent. The verb "affected" tells us it is a performance. He is like an actor playing a role, and her parents are the audience.

This is a very formal expression. It just means he tried to appear sophisticated ( showing a lot of experience and knowledge about the world and about culture, art, literature, etc.) to his parents.

Thanks 💖

Perhaps also look up "affectation". "an air of" is as when you have a subtle fragrance in the air, but usually used figuratively, a hint of, a feeling of.