Place a drop cloth under the door to protect the flooring.
Sand down the scratches. Sanding can be done by hand or by using an orbital sander. Hand sanding takes more effort, but it provides better control over how much of the wood is being sanded off.
Wipe down the sanded areas with a tack cloth to remove sand particles.
Dip a putty knife into wood filler and scoop out as much as necessary to make repairs. Apply the wood filler to the sanded areas, smoothing off the excess. Allow it to dry according to the time recommended on the label.
Sand down the wood filler until it is smooth and level with the surrounding area. Start with 220-grit sandpaper. After a few passes, change to finer 320-grit sandpaper to further smooth the wood.
Clean the sanded area with a tack cloth.
Apply paint or stain to the affected area of the door. Keep the door open until it has completely dried.