It takes a lot of leg power, balance and form to do what that video is doing (even with the right equipment) but if you look at his trail in the slo mos, you can see that it's a thin line. He doesn't "skid" his snowboard. At high enough G's in a high speed turn, the sidecut as well as the board bends making it sort of like a U-shaped ice skate. This cuts into the snow and digs a little track for this U-skate to ride along. He doesn't throw that much snow for his speed because the only snow comming out of the ground is from that tip digging out a thin "slot" along the path. If it was a skid at that speed, it would look like a tidal wave. Good carvers also end their turn by "boosting" themselves into the next turn when the board "snaps back" like a spring. It's supposed to be possible to gain speed this way and is a method racers try to perfect.
Anyways, more down to earth, you can carve bigger raduis turns without laying on the ground like them. You just have to be wary not to skid. The more you skid, the louder your boarding is because you're scraping the snow. So you can listen to yourself as you ride and can tell when you are carving by that if you don't want to stop and look up.
If you want to get "closest" to that video without hardbooting, you can use a more "forward" stance so you can square up your shoulders almost like you're facing forward like those dudes. When I was into it, I did 30 15 (or 45 30, I forget) and I had a directional sort of stiff board which helps. Then you can pretty much lean more or so "left and right" and it will turn by itself if you balance correctly. At some cases I was able to go sightly "uphill" before transitioning to a new turn too...but you have to go fast and if you slip, you might just fly into the trees in the side.