What makes me keep clinching my jaws down till my teeth hurt?

Anxiety: Anxiety is probably the most common cause.

Stress : Stress is usually the main reason for bruxing/clenching. See your dentist and have a bite plate made to avoid the problems that can occur.

Depends: The answer would depend on if you are referring to a daytime event or while you are sleeping. Daytime clenching can be a result of medications, stress, movement disorders and other factors. You should discuss this with your dentist.