The invention discloses a laser cutting machine and a cutting method thereof. The laser cutting machine comprises a supporting base. Fixed legs are fixedly connected to the periphery of the bottom of the supporting base. Sleeves are fixedly connected to the bottoms of the fixed legs. The top ends of inner cavities of the sleeves are movably connected with movable legs through springs. Wheels are movably arranged at the bottoms of the movable legs. The right end of the outer surface of the supporting base is fixedly connected with a controller. The outer surface of the controller is sequentially fixedly connected with a laser generator switch, a pressure pump switch, a sliding adjusting rotary knob and a lifting adjusting rotary knob from left to right. The top end of the supporting base is fixedly connected with a second supporting plate through supporting rods. An air filter box is fixedly connected to the middle end of the top of the second supporting plate, and the requirement of people for removing slag generated in the cutting process can be met under the action of a sponge filter layer, an activated carbon filter layer, a KDF filter layer, a pressure pump, a baffle and a pipeline, normal proceeding of cutting work is guaranteed, and meanwhile the cutting quality is improved.