Measure around the opening and use a speed square to mark a square around the area. Use a utility knife to square the edges of the opening.
Dry fit the patch by inserting it into the square hole in the wall. Adjust as necessary to achieve a good fit, then cover the wall area with drywall mud (Image 1). Insert the patch into the hole (Image 2) and press the edges into the drywall mud to adhere.
Use a joint knife or a putty knife to spread a layer of drywall compound approximately 1/8" thick (Image 1); feather the edges outward.
Cut pieces drywall tape two inches longer than patch and wider than the patch. Place the tape over the seams of your patch; rake the tape downward with a putty knife, then spread another wide layer of compound over the tape, again feathering the edges outward. Allow the area to dry, sand it smooth (Image 2) and apply paint.