Atmosphere is made up of gases that we collectively call “air”. These are gaseous molecules that are very loosely bound to anything or even with each other. Since nothing is pulling or pushing them, they are free to move. But, of course there is gravity acting on each of them pulling them towards earth. Why don’t all of them get pulled down, crush us and form a layer near the ground? The reason is that, each particle has thermal energy (heat) that keeps them moving in random directions, up or down, left or right, and forward or backward. These thermal forces that make them float are stronger than the effect of gravity.

So, instead of being pulled in one direction alone by gravity, they float. So all of the “weight” of our thin atmosphere does not act on us (thankfully), preventing the collapse of frail beings likes us.

Here is another way to think about it: if the weight of the atmosphere did have an effect, we or a significantly different being would have evolved to accommodate this (maybe crawling) if life was still possible.