How can I repair damaged door casing?
I would just replace the trim. If you were good with an oscillating saw or sharp wood chisel you could cut the molding just above the damaged area at an angle. Remove everything from there down.

When you take out the lower piece try to keep a piece that is undamaged so you can take it with you when you go searching for replacement trim. Hard to tell from the picture just how standard that trim profile is and when you go to the store it will be very handy to have a reference. 

Cut and fit the new trim to fit the angle and length to the bottom and then nail in place with small finishing nails. A small amount of spackle can be used to blend the joint if it is not a perfect fit.