An airy place is full of fresh air. Other airy things are either light like air or without substance like air.
Air surrounds us, and it has a lot of meanings. If a place is described as airy, that means it's full of fresh air. An open stadium is airy, while a stuffy apartment is not. If a house has lots of open windows, it's airy. Also, an argument that has no substance can be called airy, because there's nothing to it. Light things are airy too: a flimsy, light dress or a low-fat dessert could both be called airy.