What would cause extreme pain in both hands?

It would depend if it just happened and has only gone on for a few hours vs a longer period of time. For the former, try taking a benedryl every four hours and 15 minutes with a towel-covered ice pack. If the problem persists more than a couple of hours, check in with the doctor. For the former--continuously painful hands and swelling, get to your doctor--you might ask about getting blood work for Arthritis or another autoimmune illness.