You could make an analogy between the pressure distribution of a sound wave and the mass density distribution of a realistic spring undergoing vibrations, but it wouldn't give you the explanation you're looking for. As a matter of fact, that would be more like explaining a sound wave in terms of springs, rather than what you're trying to do, i.e. explaining a spring in terms of waves.
Although I'm not intimately familiar with the details, basically what goes on at the microscopic level of a spring is that, when the spring is at equilibrium, the atoms are set in some sort of rigid structure. Any given pair of atoms has a potential energy which is a function of the distance between those two atoms, so the entire spring has a potential energy determined by all the distances between every possible pair of atoms:
$$U = \sum_{i,j} U_{ij}(r_{ij})$$
In equilibrium, the spring will take a shape which minimizes this total potential energy.
If you think about it, a metal spring might typically be formed by heating some metal to make it malleable (or even melting it), and then forming it into the desired shape before it cools. The heat allows the atoms to move around relatively freely so that they can reach the equilibrium configuration that minimizes their potential energy, then once the spring cools, they are frozen in place.
Of course, the atoms are not completely frozen in place. As I see that Georg has already written in his answer, the potential energy between two atoms ($U_{ij}(r_{ij})$) has a minimum at their equilibrium distance and goes up on either side. If you add some energy into the system, say by exerting a force on it, you can get the atoms to move closer together or further apart. When you stretch or compress a string, you are really just doing this to all the (pairs of) atoms in the spring simultaneously. The atoms will, of course, "try" to return to their equilibrium position, i.e. they will "try" to minimize their potential energy, and this is what you feel as the restoring force of a spring under tension.
Good question! What you probably haven't been told is that forces can sometimes go by multiple names. There can be a name that describes what produces the force, but also a name that describes how the force is acting, or something else. The term "restoring force" falls in this latter category: it's a name that describes which way the force points, i.e. toward the equilibrium point. The same force might also have another name which describes what produces it. For example, it's quite possible that the tension of a string, or the elastic force of a spring, or gravity, is the restoring force.
In fact, it's possible that multiple forces contribute to the restoring force. "Restoring force" just means the total force that acts toward the equilibrium position. In the case of the bungee cord, the restoring force is the sum of gravity and the elastic force of the cord.
Another case where you might have seen the same thing is with the term "centripetal force," which refers to whatever forces are pointing toward the center of an object's circular motion. Sometimes gravity is the centripetal force, sometimes it's tension, sometimes it's a normal force, etc., or even a combination of different forces.
Best Answer
Yes, the forces involved are interatomic (and so fundamentally electromagnetic) in the case of a stretched spring, as for a stretched wire or a stretched rope.
I think, though, that the passage you quote is misleading. If you formed a helical spring out of a metre of steel wire, then the extension of the spring when subjected to equal and opposite forces at either end would be much greater than the extension of the original wire when subjected to the same pair of forces. This is because, for the spring, there is twisting and bending of the wire. Extension of the wire along its length is negligible by comparison. The shape of the spring is indeed magical!