Last winter I got to try this really awesome "toy" at Bristol Mountain: a giant airbag jump done by Acrobag (http://acrobag.org/). For an hour I could go off the big jump, do anything and wanted, and not get hurt if I messed up a landing.
This reason for this is completely related to what we are doing in physics this week. The change in momentum (impulse) is over a longer period of time which makes the force your body feels much less. For example, for proof of concept we can assume that when a person hits this bag they are in free fall so their velocity = sqrt(2gh). So they are going at about v=sqrt(2*9.8*3)=7.7 m/s. Therefore, their momentum is about p=(70kg)(7.7m/s) = 539 kg*m/s. We can use impulse to find the force they would feel without the Acrobag. F=(change in p)/(change in time) or F=539/.001=539,000 N which would definitely cause some injury. With the Acrobag, this force is greatly reduced. F=539/2=269.5 N...Obviously a huge difference in the force the person feels.