thin air does not hold them up

I cannot understand the bold section of following sentence. Please help me.

It is hard for birds to breathe in thin air and thin air does not hold them up.

The wings of birds are there to prevent them from falling down: the wings rest on the air, so that the air supports them, it holds them up, it prevents them from going down. This is part of the mechanism that birds use to counter gravity. If the air is too thin, it cannot serve the wings for support as well as thick air can, it cannot hold them up, they will fall down.