All solid materials are made up of atoms which interact with each other by means of a complex variety of forces. You may visualize these interactions as shown in the figure below. In the diagram, the yellow circles represent atoms and the springs represent the forces between the atoms.
The figure on the left shows a top and a side view of $NORMAL-AGENT$'s surface when $BODY$ is not on it.
The figure on the right shows a top and a side view, of $NORMAL-AGENT$'s surface when $BODY$ is on it. $BODY$ compresses the surface molecules of $NORMAL-AGENT$. The compressed molecules of $NORMAL-AGENT$ push back on $BODY$, exerting a force perpendicular to the $NORMAL-AGENT$. This is the normal force, exerted on $BODY$ by $NORMAL-AGENT$.