Step 1: prep the hole
Step 2: Brace the backing
Step 3: apply patch
Step 4: Mud and tape!
Step 1: Prepping the Hole
1: cut your hole square, in the picture this is where for some unknown reason the cold air intake in my newly purchased house was routed.... making the entire downstairs freeze in the Canadian weather.
you may notice that this is on a unfinished wall, if this were a finished wall i would lightly sand the paint work around the cut as well to make it ready for the mudding and taping that is to come.
Step 2: Bracing the Backing
1: size a block to be put into the hole, slightly larger than the hole height wise, 2X4 is probally the best as most frames in houses are done with these, although apparently a few are done with 2X6's.
2: put the board into the hole and turn in to be snug in the space in the wall, if insulation is present just push it out of the way for now and once the board is secured pull the insulation tight to the board to ensure a good insulation.
3: Attach with drywall screws at each end,
Step 3: Finish It Up
then after this you will mud and tape the edges of it.
if on a painted wall i suggested earlier to sand the paint a bit out from the hole lightly, this will help merge the paint into the hole and not make it as noticeable.
And that is the end. fairly simple!