gravity - How does weight add up to press on things?

I think I understand how pressure works with gases. More molecules bouncing around -> more random impacts -> stronger force.

But I realized to my embarrassment that I don't understand how solid things press on each other, on a molecular level. Say I put a block of iron on my head. If I put another one on top of it, I feel twice the weight. The two blocks together can tear through thin paper where one block can't. But the contact between my head (or paper) and the blocks is just a very thin layer of atoms of the lower block's structure. If the lower block doesn't move when I put the upper one on it, what causes this thin layer to "press" on my head (or paper) more? When the two blocks together tear through thin paper, where does the force come from that acts on the paper molecules - it can't be gravity from the upper block, right? And how come that whatever this source of pressure is only depends on the weight of the upper block, and not on what it's made out of, iron or wood?