Patching drywall is not as hard as it might seem. You just need the right steps!

Use a drywall knife to cut out the damaged section of the wall. Cut a nice even square and pull it away from the wall.

Make backers for your patch by cutting some one-by-two (1x2) boards so that they are a couple inches longer than the hole.

Insert one of the backers through the hole and hold it flush against the inside of the wall, on the top of the hole. Use a drywall screw at each end to secure the board to the wall. Then repeat the process for the bottom of the hole.

Cut your patch from a scrap of drywall using your damaged section as a template if possible.

Carefully tip the patch into place so it fits securely in the hole, and use some more drywall screws to secure it to the backers.

Use some self sticking fiberglass tape to seal the seams of the patch. Then cover the tape with spackle. Allow it to dry and then sand the surface. Apply two more coats of spackle, sanding it after each has dried for a flawless finish.