What exactly is a surgical drain?

Nothing to worry about. If the antibiotics don't appear to be working the doctor will insert a tube into the wound to drain off the pus and lymph etc. to reduce the swelling, and get rid of the breeding ground for bacteria. It won't hurt because he'll inject a local anesthetic before doing the procedure. Animal bites (especially cats) are particularly dangerous because their teeth act as syringes, pushing the bacteria from their mouth deep into the skin.