Get a fine nail or brad, and nail it in on the corner so that the head does NOT stick out past any of the surfaces. Not the top. Not the front. Not the side. Use a straight edge to make sure.
I'd say don't sand down the wood for a smooth surface. When it's frayed like that, it would give more surface area for the wood filler to bind to. Use wood filler to fill it in, the nail / brad will act as structural support and more binding to hold it on there. Sand the dried filler to match the size of the missing wood.
Then you have to get artistic. Paint it the background color. (If you get lucky and work at it, the wood filler can match the background wood color.) Then hand paint on the grain lines.