This is what my intuition says about how the concept of weight works in a solid.
The light brown arrows are the vectors of gravitational force, the light gray parts are the electron clouds around atoms, and the red arrows are repulsive forces. The gravitational force pulls on both the rows of atoms. The first row pushes on the surface beneath it with a force equivalent to $mg$. There is another layer of atoms above it which is also being pulled by the gravitational force.
These atoms tend to move towards the force, but cannot as there are electron clouds of other atoms beneath them. (Electrons repel each other). So, they exert a force equal to mg (on the atoms beneath them). This results in the adding up of the forces and the total force exerted (by each column) on the surface becomes $2mg$ (assuming all the atoms are of the same mass).
Also, my idea of how this works might be totally wrong. I have just imagined and assumed all these things with a very few facts I've known.
Please do correct me if I am wrong.
I usually love to learn everything at the level of particles, please help me understand this concept the same way.