Why do lasers cut? Is this a case of light acting as matter?

When lasers cut something, they're only cutting in the sense that they're making atoms be not as attracted as they once were to each other. When you get down to the nitty-gritty details, it is not really the same as mechanical cutting.

Remember that lasers shoot photons, and when photons hit atoms, they excite electrons. If you excite these electrons enough, they'll have enough energy to disassociate from the atoms they formerly "belonged" to. This makes individual atoms disassociate with whatever other atoms they were once bonded to, and in the mad scramble to go to a lower energy state, they very likely do not go into the same configuration they were before. Some atoms, like the ones directly hit by the laser beam, go to a vapor and float away. Others "choose" one side of the material to go to. Any bonds the material had with itself is then dissolved, so it is effectively cut.

This is different than, say, taking shears or scissors to the material. The methods of those things cutting are purely mechanical, and you don't have to worry about vapors as much as when cutting with a laser. (You also don't have to worry about reflections from materials, either!)

Cutting is a process when you deliver energy to break chemical bonds in material that you cut.

When you use a saw, you deliver mechanical (kinetic) energy that converts into kinetic energy of particles of the thing you cut, so they can get out of the thing.

Laser is just another way to deliver such energy, since the a photon has enough energy to break some bonds and deliver some heat for the molecules that can evaporate.

Since cutting is just breaking chemical bonds and removing particles in some specific place, laser has ability to cut. It has not much to do with newtonian "light particles" approach.

Think of the laser process as being similar to melting a substance through a change in state.

An analogy might be putting a hot wire on top of an ice cube. It makes a 'slice' by heat, not by cutting. It turns the solid state ice into water and gas which doesn't hold together the same anymore.