Why do lasers cut? Is this a case of light acting as matter? All I found in Google was very broad. From a physics models perspective, why can photons emitted from a laser cut? Does this cut mean that the photons are acting like matter?
 A: 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.
A: 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!)
A: Photons posses momentum and are affected by gravity. So, they act as matter all the time. To use the effective momentum for laser cutting requires very high photon density which isn't practically feasible.
Industrial laser cutting uses extreme heat density to cut. There are many methods viz. vaporization cutting, melt & blow, thermal stress cracking, etc. See Wikipedia page Laser cutting, section Methods.
A: Regarding as how the laser light can deliver such energy to cut something, it is because it is "coherent light". An old example to imagine this is, imagine a troop of soldiers over a bridge. If they march (i.e. everyone stomping synchronously at the same pace in the ground), they will add up each stomp at the same time, adding up energy as if all the troop were a single giant boot stomping at the bridge, so it could even break and fall - that´s why groups of people (should) never march over a bridge.
Each photon in the laser is synchronously coherent with each other, adding up energy to the beam instead of scattering the energy each on its own as a common lamp do. So the beam will be so intense over a small region of matter to the point of delivering energy to it so it breaks (burns) apart. 
A: 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.
A: Lasers "cut" as an interaction between matter and radiation (themselves different forms of natural substance)
A LASER (Light Amplification by Stimulated Emission of (Coherent) Radiation ) beam being coherent and high-energy (quantum-mechanically a-la Einstein-Planck relation, energy is directly analogous to laser frequency) when interacting with atoms and molecules, provides enough energy for electrons and molecules to move to higher energy-levels and underlying bonds become loose or break completely, resulting in the macroscopic result of "cutting".
As noted above only certain types of LASER "cut" certain types of matter compounds (related to frequency/energy of beam and the bonds base energy of the matter compound under study)
