A clinch cutter is essentially a heavy steel chisel with a handle. When the blade is positioned under a horse shoe nail clinch and hit with a hammer, it cuts or bends out the clinch, allowing the nail to be removed.
This is the most effective way to remove a clinch and thus a horseshoe nail without damaging the hoof wall. A rasp can also be used to remove clinches but takes much longer and may thin the hoof wall if used incorrectly. Grinding of clinches with a rasp also dulls the teeth of the rasp.
Since you will only use the tool rarely, it is fine to use a lower quality, lower cost model. Your farrier or veterinarian should teach you the proper way to use the tool.