Where does the expression “to give the air to a person” come from and What does it mean?

The expression “to give the air to a person” means: To tie the can to; to fire; to dismiss; to discharge from employment; literally, to put out of doors.

A British counterpart of this modern American slang is “to give one the sack”.

The expression also has a negative meaning in the language of love.

When Joan gives the air to John, she terminates a courtship by giving John to understand definitely and unmistakably that his attentions are unwanted, or she breaks off an acquaintance before John can begin to get romantic notions.