Poke through the drywall under the cabinet with a scratch awl in several locations. Insert the blade of a drywall saw into one hole.
Cut out the damaged drywall using the drywall knife, making a square or rectangular hole that can easily be filled with a same shape and sized drywall patch. With the old drywall removed, you will see the wall studs to which the drywall attaches.
Measure the square or rectangular opening with a tape measure, and then transfer the measurements and mark cut lines with a pencil on a sheet of new drywall.
Cut the drywall to fit the rectangle or square under the cabinet with a drywall saw. Fit the patch into the square or rectangle.
Fasten the patch to the studs under the cabinets with drywall screws and a screw gun. Then apply drywall tape atop the seams of the patch, centering the drywall tape over the seam lines.
Use a putty knife to cover the tape with joint compound. Allow the joint compound to dry and then sand it smooth using medium-grit sandpaper.
Wipe down the drywall patch with a tack cloth, and then vacuum under the patch with a hand vacuum to pick up all the dust from sanding. If desired, texture the drywall in the same manner as found on nearby walls using the joint compound. Lightly sand when dry, and then paint the drywall patch with a matching paint using a paintbrush to finish the repair.