There are 3 main innovations in this way of skiing.
1) skiing at terminal velocity - you aren't trying to slow yourself down. You are more concerned about a route down, which if involves rapid turns, will convert forward speed into cross-run movement anyhow, so forward speed is reduced anyway without specifically trying. It's more like the feeling of a rollercoaster with a track route you create in realtime - to me it's an evolution of skiing and feels completely different and is exhilarating.
2) turning by relaxing - if you're in a car going around a bend and you let go of the steering wheel, the steering wheel will spin automatically to come out of the turn and then move into an opposite turn to some degree which you can then reinforce by gripping and continuing the steering again. It's the same principle with heartcarving: you want to switch from one turn to the next? Just relax the legs and the heart will move to the opposite side of the feet. Want to start turning when in a straight line? Just lift a leg and you'll lean over and start carving. Place pressure on the foot again when at the desired body angle. In the videos it looks like I am moving my legs under me, I am not, I am relaxing my legs and my upper body is moving over to the opposite side of my feet, with only an appearance of moving the feet from side to side as with more conventional skiing. (however I can consciously move and swing my feet as a throwback to a more conventional style of skiing - good for moguls - so you still need that skill)
3) Skiing from the heart - as an exercise, the body is split into two parts, heart and feet. And then one can think about the heart as a sphere which can move in 3 planes: pitch, yaw and roll. Also, one isn't using poles, instead they are playing with centripetal acceleration via relaxation and tension in the legs and more pronounced movements of the back (which affects the heart's pitch, yaw and roll.) The most basic heartcarves can be done on a beginner's run with very low bodily effort or tension, or pressure (see lower video.) At higher speeds and more difficult runs, it is more intense and fun.