Friction mechanisms vary depending on the materials, but surface destruction is not necessarily required. When a box is on the floor new bonds form between the atoms of the box and the atoms in the floor. These bonds between the objects are generally a lot weaker than the chemical bonds holding the box together, but strong enough to be noticeable when there are $\sim10^{23}$ of them.
You can look at friction from a force point of view or an energy point of view. In terms of energy, the energy required to break the temporary friction bonds comes from the box's kinetic energy and becomes thermal energy in the box and floor.
In terms of force, when the box slides across the floor, the temporary friction bonds apply tiny but numerous forces that tend to slow the box. These forces pull equally and oppositely on the floor's atoms, but of course the floor doesn't move. Instead the floor's atoms wiggle more than they were wiggling before, which is a way of saying the floor gets warmer. The same happens to the box's atoms, so the box gets warmer too.
So far I've talked about "kinetic friction", that is sliding friction, but consider the "static friction" that occurs when you push on the box weakly so that it doesn't slide. Nothing is being broken, yet friction is still happening. In this case those temporary friction bonds are holding, providing enough force to counter your push. Only when your push force exceeds the bond strength do the bonds break and the box begin to slide.
