Answer:
It's all about energy transfer, and the rate of energy transfer. If you hit your finger with a hammer, the pain is proportional to the square of the speed at which you move the hammer. You are giving the hammer kinetic energy by accelerating it from rest. The key is 1/2mv² where m is the mass of the hammer and v is its velocity.
Scale that up to two cars crashing and can see that the crash damage is proportional to the speed and mass of the cars. Obvious, but it is important to remember the damage is proportional to the SQUARE of the speed.
Cars are designed to crumple in an impact so that much of the energy of the impact is dissipated in bending steel - ultimately turning into heat energy. But there comes a point when the steel can bend no more - it's fully crushed - and so the car and its passengers decelerate much more quickly, and humans are very vulnerable to deceleration. So you need to reduce sharp deceleration with local energy absorbers such as seatbelts and, better, airbags.
