Question

How do airbags reduce damage to passengers during an accident?

Answer 1

There's two main concepts here: impulse (change in momentum) and pressure (force over some area).

Impulse
FΔt=mΔv≡I

In a car crash, you decelerate from some initial velocity which you were driving at to rest. Regardless of any specifics of the crash, if you start with some vivi, your change in momentum will be the same. So our goal to reduce the amount of force acting on you is to increase the time over which this deceleration occurs.

If instead of an airbag, there was a brickbag. Your Δt would be ridiculously small, which makes your F ridiculously large. If you were in a crazy hospital room, would you want to run into a wall of bricks or a wall of pillows? Exactly. [unless you're truly crazy]

Pressure
P=FA

Pressure is "force per unit area". You wonder why not only do we use an airbag which is cushion-y and absorbs a lot of the impact to increase the duration of deceleration, but why is it SO BIG (large surface area)? Let's say you're a fancy engineer and you've designed the most perfect cushion - it just happens to be shaped like a nail. Well, you've done your best to minimize that force (see impulse section above) but that surface area over which the force applied is so small that the pressure is pretty much arbitrarily large! Congratulations, you have an airbag which stabs its survivors!

To reduce the impact which an airbag has on its passenger, it should have a large surface area to distribute that force over a person's body to minimize the injuries sustained.

Random Fact
Airbags also come with vents so that when you hit it hard, some air gets expelled to reduce the force even more and increase the duration of the impact.