When we say "air of" it's like we're describing a perfume or odor that the person is giving off. Sometimes you approach someone and you can sense something about them. There is something that your sense of smell is telling you... "ahhh she is wearing a perfume". So there is something in the "air" around her. But when we say "she had a slight air of indifference", we don't mean that you can "smell" indifference. It's just a poetic way of saying there is something that your senses are detecting JUST LIKE your sense of smell can detect a perfume in the air. "He had a slight air of superiority" - means your senses are telling you that he feels superior to the people around him. "The team has a slight air of confidence" - means you can feel how the team feels confident about themselves. "She has a slight air of mystery about her" - means that she seems mysterious to you, but not in a way that you can easily describe. If I say "she is mysterious" then this is a definite statement, but if I say "she has a slight air of mystery" then I am saying that this feeling of mystery is very subtle and not immediately obvious. Other examples... "with a slight air of caution, they opened the door" "the police approached the car with an air of authority"

He has a slight air of indifference. Sometimes he gives the impression that nothing really bothers him or some things are not important. But in reality he worries about everything.

With a slight air of indifference. That would mean giving a slight impression of indifference.