Remove any splintered or broken pieces from the door to get a neater area to work with. Pull off or cut off large splinters. Sand the area with rough sandpaper and dust with a tack cloth.
Scoop wood filler out of its container and onto a putty knife. Smooth it into the crack or broken area. For deep cracks, layer the filler into the hole to reconstruct a solid piece of wood. Use the blade of the putty knife to smooth the filler so it is flush with the door.
Allow the filler to dry according to package instructions -- at least 24 hours.
Sand the area with 100-grit sandpaper and dust with a tack cloth. Ensure the patched area is even with the remaining wood's surface.
Finish the repair by repainting the wood. Use a water-based primer on the door followed by a water-based paint. Allow each coat to cure before applying the following with the next layer. When finished, you won't be able to see the repaired area.