When a vehicle crashes, it decelerates very quickly. A large deceleration means a large force on the people inside, because force = mass × acceleration. Large forces cause injuries such as broken bones, damage to organs and head injuries.
The size of the deceleration depends on how quickly the speed drops. Deceleration = change in velocity / time. A car at 20 m/s that stops in 0.1 s has a deceleration of 200 m/s2. A 60 kg passenger would then experience a force of 12000 N. If the same car took 0.4 s to stop, the deceleration would be only 50 m/s2, and the force four times smaller.
Safety features work by making the stopping time longer. Crumple zones at the front and rear of a car crush in a collision, which lengthens the collision time. Seat belts stretch slightly, and air bags and cycle helmets cushion the body, so the force on the person is smaller.