Many recent recipes I'm seeing for bread, this one for example, are using a strech and fold technique rather than a traditional knead. Whats the advantage here? It sure seems to work, but I'm not sure why it works well with so much less work.
The purpose of all kneading techniques is the same: as @rumtscho wrote, to stretch and align strands of gluten.
The traditional kneading method is to fold the mass of dough on itself, press it together with the palm of the hand, turn, and repeat.
With very high hydration doughs this is impossible because the dough will hopelessly adhere to the hands. The technique is, therefore, modified to reduce hand contact with the dough as much as possible.
A dough scraper is used to gather and fold the dough which is laid over on itself instead of being pressed. The result is the same as normal kneading but only the fingertips come in contact with the dough. After just a few turns the gluten sheets will align and the dough will still be tender but much less sticky.
As for why it seems more effective I can only conjecture. Gluten forms in the presence of water. Higher hydration doughs allow for faster gluten development. It is not that the folding is more effective but that kneading very wet doughs progresses faster.
Stretch-and-fold often works better with less effort because it is done periodically (as in the recipe linked in the question, where you do a stretch-and-fold every 45 minutes over a few hours).
The molecules that make up gluten strands take a certain period of time to align. By waiting for the gluten to form nice networks of strands, you can then stretch them to their greatest extent, which will add strength to the dough. By waiting a period of time again, you allow even more bonds to form, which you then stretch out and extend to add strength.
Compare this to traditional kneading, where you mix the dough and then pound, stretch, and fold for 5-15 minutes immediately. The gluten has barely been hydrated and it's only beginning to form networks. That's why a lot of recipes recommend an autolyze strategy, where you wait at least 15 minutes after mixing the liquid before kneading. By doing so, you allow the gluten to hydrate more fully, and you generally cut your kneading time in half (or even less).
Periodic stretch-and-folds just act like repeated versions of this autolyze strategy. In most bread recipes, you can generally get away with avoiding kneading altogether -- just mix until all the ingredients are evenly distributed and there's no dry flour. Then, just do 2-4 stretch-and-folds spread out during the first rise. The required number will depend on the type of dough.
Stretch-and-folds actually are a great way to monitor your dough during the first rise too. You can periodically check the texture and make sure it's actually developing enough elasticity -- if not, add in another stretch-and-fold at some point. If the dough is getting too strong (sometimes if the gluten network gets too tight, it can actually hold back the final proof and oven spring), do fewer.
The "stretch then wait" strategy is also similar to the "bench rest" often done before final shaping. Again, many recipes recommend doing a few stretches and preshaping maneuvers after the first rise, then letting your dough rest for 10-20 minutes before doing the final shaping. Essentially, you're doing a stretch-and-fold of the gluten, then letting it "relax," and then stretching it again once even more connections have formed. If you're not doing that before final shaping, you're probably not getting the highest rise you could. Doing it periodically over the first rise will add a huge amount of extra strength to the dough -- generally much more than you'd get even with an extensive initial knead.
One other technical note: initial kneading (and particularly mechanical mixing) introduces oxygen into the dough, which will actually contribute to the destruction of some flavor components in the final bread. It's a relatively minor effect, but substituting stretch-and-fold maneuvers instead of a long knead will introduce less oxygen into the dough initially, which can make a slight flavor improvement on the final loaf.
As other answers have mentioned, with really sticky high-hydration doughs, stretch-and-folds are often the only way to "knead" the dough. But there's really no reason why not to do stretch-and-folds with most bread doughs during the first rise. I frequently do this rather than kneading a lot at the outset (which, frankly, isn't really efficient), particularly with tough heavy doughs like whole-grain or multigrain breads.
First of all, a traditional knead is stretch-and-fold. I don't know what you were taught, but the technique my grandma taught me is:
While this uses less force than the video you linked, and makes smaller folds, it is still folding.
Second, you asked why. This being dough, it is all about gluten.
What you want to achieve is to unfold the proteins forming the gluten and orient them in the same direction, making them parallel to each other and creating lots of end-to-end bonds (as opposed to cross bonds). The nice side effect is that you aerate the dough, trapping miniature air bubbles into it. This results in a fine, even crumb.
There is a beautiful illustration of this in On Food and Cooking. One look at it, and you know why you should stretch and fold.