Can I just cut this part of the door frame out and replace it?
I think the easiest thing to do would be to remove the doors and sand it down so that it is all flush and even.  A belt sander would make quick work of this it can be done by hand too if you're willing to put in some sweat.  After sanding it, chisel out material for the hinges, prime, paint and re-hang the door.

It's hard to tell for sure but it looks like the hinges are recessed too far into the door which might be why it didn't close properly.  The hinge should be flush with the edge of the door.  Place a couple washers behind it to bring the hinge out flush with the door.