Once the support piece is sufficiently secured (whether you glue it or screw it) you're ready to fit a piece of drywall for the hole. Make sure that you are using the same thickness of drywall that is already installed. Cut the piece so that it is a fairly good match. You don't want 1/4" gaps all around. An 1/8" is MAX.

Now glue or screw the patch piece in place. This depends on how big the piece is (and if your backer is solid oak). If the piece is too small, I'd say about the size of this patch (6" or less), you'll risk busting the patch when you screw it so glueing is probably the best option there. Rasping the edges is not necessary, provided they're not very jagged. You can cut the piece with a utility knife (from scrap if you are fortunate) or use a drywall saw.

Preparing For Mud

Something that I do when working on a project like this, is score the paint and paper around the hole and then peel it off. You'll notice in the image that I only go out a couple of inches. This will be filled with a thin coat of mud and will allow me to taper the mud and provide a finish that blends in better.

Once the adhesive is dry (or if you screwed) you're ready to put your first coat of mud and tape on.