Summarize:

Use 320-grit sandpaper to sand down any areas where the paint is peeling and/or the wood is splintering. Clean up the scratched area so that you can patch it rather than piecing it back together. If the wood is significantly splintered, wear heavy gloves while sanding to avoid injuring yourself. Squeeze or spread a small dollop of wood filler over each scratch on your door. Use your finger or a putty knife to spread the filler around and press it into the scratches. Try to achieve a smooth finish that is even and level with the rest of the door. Wood filler is available online or from your local hardware store. It will either come premixed in a tube or in 2 separate parts that need to be combined before using it. Follow the instructions on your wood filler for the best results. The filler needs to set up and fully adhere to the wood in order for it to repair the door. Once you have pressed filler into all of the scratches, leave it for 15 to 20 minutes to dry fully, until it is completely solid to the touch. Check the instructions on your wood filler for more information on expected drying times. Once the filler has dried, use very fine grit sandpaper to sand down previously scratched area. Work in smooth, deliberate strokes to remove any excess wood filler and flatten the surface of the door. Painting over the sanded area will make any irregular bumps more obvious. Run your hand over the area you patched with wood filler to catch any areas that need more sanding and smoothing. Sanding the wood and paint around the scratched area can produce a significant amount of dust and debris that can ruin the look of the repaired door. Dampen a clean cloth or a paper towel and lightly wipe down the area to clean up any dust left behind.
Use very fine grit sandpaper to remove loose splinters and paint. Apply a generous coat of wood filler to the scratched area. Leave the filler to dry for 15 to 20 minutes. Sand down the wood filler with 320-grit sandpaper. Clean the area with a damp paper towel or sponge.