Looks like its just a roll cast followed by a few false casts to get the line out to the required length. With the small water they are fishing it looks like he has very little fly line out - primarily casting the leader.
It looked to me like a very short - what i call the oval cast or circle cast . i practice this cast by casting forward off my right shoulder
and bringing my back cast around and over my left shoulder forming an oval , or circle . he just did it with the tip of his rod .
most the time making this cast at distance i will drop the rod to 3/4 angle like throwing a ball 3/4 arm instead straight of over hand to make the circle clear around yourself , or oval shape when you get a lot of line in the air . after you get that down drop the rod level to the ground for a side arm under the brush cast . its funny how gravity seems to not play as much with you loop when doing this and the cast can be pretty simple and far - the old side arm !!!
It's not something that is specifically taught. All he is doing is getting more line out of the end of his rod. If you are trying to feed more line out when you first approach the water you will naturally do this. This is also done right after you retrieve your line back after landing a fish or changing your fly. I think the best instructor for teaching this is getting out on the water.
If i did this right - joan wulff does this cast on a larger scale at 1:23 in the youtube video . i would suggest practice in the yard or the park , i'm teaching two friends right now and they wait tell on the water to practice which is a pain ! if your going to be good at any sport it takes practice . but thats just coming from a baseball coach and fast pitch softball . why wait to learn on the water and make it more difficult for everyone else around you . take the time to take your casting to a different level on your own , or hire a instructor . you'd be amazed how fast you can learn and get better with practice .
Deepblue, you're on track. Check out a local fly fishing club; most provide group lessons periodically, from beginner to advanced, for a very reasonable price as a fund raiser and to find new members. Additionally, most local fly shops schedule basic fly fishing classes that include casting lessons. Both are great places to start. Classes are cheaper than a private instructor, save that for later when you can take advantage of the one-on-one instruction. The guy in your video wasn't doing anything particularly note worthy other than keeping his back cast high and out of the bushes while false casting to get a little line out.