Why is the layers often called a cozy blanket around earth?

The entire atmosphere is considered the blanket for the earth because of the protection it provides from solar radiation. This includes gamma rays, X rays, and severe ultraviolet light. The way it works is as follows:

Nitrogen in the atmosphere breaks down when exposed to gamma rays, and in the process the gamma rays are dissipated. Oxygen breaks down when exposed to X rays. And believe it or not, Ozone protects us from ultraviolet rays using the same process.

Also, when an access amount of solar radiation hits the atmosphere the atoms get excited, and this effect causes the Aurora Borealis.