Why do cars shimmy when they stop?

If the car shakes when stopping you probably have warp brake rotors (discs). Have a trusted mechanic inspect those brakes for safety sake.

Define shimmy. Heavy braking in adverse weather/road conditions will cause the ABS to operate. The ABS works by alternately applying the brakes almost to lock-up and then releasing them. If the shimmy only happens in adverse weather conditions it could be that the ABS needs looking at but it's more likely that you probably need to improve your driving skills.