YouTube - tumbling
There are some good examples in there. Look for when he doesn't get the stick and actually falls. The way most people fall is the incorrect way, where you sit down and hold out your arms. This is bad because it minimizes the points of contacts, so all the force goes into those points, that's when you get bruises and broken bones.
Tumbling is when you find that balance between spreading yourself out such that the force acting on a specific spot is minimized by increasing the surface area, and having a slight curve so that you can roll out and bleed off that momentum.
When I fall, I do something that resembles that. Not sure how well it'll work with a helmet on. I'm very accustomed to falling on my head. Have done it a ton of times on surfaces harder than hard packed snow (haven't gone on ice), so when it comes to snowboarding, I don't have a problem with the surface I'm tumbling on. I taught myself this past summer since I needed a way to bail out of my headstand to handstand presses, and particularly when I take it to the rings, go inverted, and my grip fails for whatever reason.
On the ground, where you're not moving, and you fall on your head or butt, you have enough momentum to stand up. So you can imagine while snowboarding you'll have more than enough momentum, so you just have to figure out how to turn that momentum into kinetic energy without overshooting. Or you could just bleed off that momentum on the roll out, stand back up, and then continue. Not as cool, but it still won't hurt as much as a regular fall.
That guy's real lucky he gets to do it on ultra soft mats. I get to do it on rocks!
I'm not sure how many snowboarders tumble. I know in my group of friends I'm the only one, so snowboarders may not be the best people to ask on how to fall properly. You could always ask gymnasts, parkourists? (people who do parkour/free running), bboys/bgirls (breakdancers), and all those types of people.