They want it in case your auto insurance denies their claim. Also, since you were the third party (not at fault in the accident), your insurance company may want to wait until your claim is settled before they pay. The chiro won't want to wait that long and may bill your health insurance so they can get their money. Your health insurer will then be paid back by the person who hit you (or his insurance carrier). It's called "subrogation" if you want to investigoogle it further.
Thanks a lot for the clarification jhunter, that does make a lot of sense. Given the rest of the hogwash they were trying to force on me, I just got a little paranoid about it.
I'm also surprised no one commented on the video camera in the room where they had me changing my clothes. That simply must be some sort of violation of privacy? To ask you to undress in a room with a camera and not even mention the fact the camera was there and pointing right at you?