You can rent floor buffers from rental shops or home centers. Be sure to get medium and fine screens. Remove all the furniture from the room. Place the medium screen on the buffer, turn it on, and push it back and forth in the direction of the floorboards across the entire floor. Your goal is to grind off the top layer of the gloss. Do not stop the sander on the floor while it's still running. That can damage your floor. Make slow turns and don't muscle the buffer across the floor. Both could leave gouges. When finished, vacuum up the dust. Use a tack cloth to pick up the dust that the vacuum misses.
Replace the medium screen with the fine screen. Buff the floor in a back and forth motion, going with the floorboards. You'll want to completely remove the scratches with the fine screen. Vacuum the dust completely. You should see a dull, clean surface where the cat scratches used to be. The stain on the wood should remain intact, so be careful not to over buff.
Once all the dust is gone, brush on your varnish in the direction of the floor boards. Do not apply too much varnish. Coat the entire floor and let it dry completely, at least 8 hours. A longer drying time may be required, depending on your area's humidity. Consider using more durable varnish. There are brands more suitable for high traffic areas.
Once the first coat is dry, sand the surface using fine sandpaper and a pole sander. Use quick, light strokes. You're just trying to dull the glossy surface. Pick up the dust with a vacuum and tack cloth, then brush on your second coat of varnish in the same manner as the first coat. Again, don't apply it too thick. Let it dry for 24 hours before moving any furniture back into the room. If the cat scratches were confined to certain areas, you might consider placing decorative rugs in those spots.