I mix dough in my kitchenAide all the time, and this happens all the time, so through the process I check the consistency and dryness and I've developed a feel for the "just right formula", but being ADHD I don't go in with precise measurements each time, but that would be a good idea once perfection is found. So anywho, I take a strong spatula from time to time and cut it of the hook, to give it a nice restart with hook on outside of dough ball. but your dough needs to be dry enough that it is cleaning itself off the bowl and it will the hook too. It's designed to find that perfect balance so that it starts wet and sticky and then as the gluten starts to web it becomes self cleaning, and the kneading is drying it a little or maybe the material is absorbing the liquid, either way it goes from dry to wet. Another thing I do if the hook is wrapped in dough instead of working against the sides, I briefly turn up the speed and that will throw the dough to the sides and sometimes clean it.
The reason for doing delayed fermentation in the fridge is that the yeast development is slowed down, while still allowing the enzymes that naturally occur in the flour to do their work (converting starches to sugars, making a more flavorful dough).
The risk of doing it on the counter instead of the fridge is that the enzymes are working AND the yeast is working, which can over-leaven the bread, and they'll eat up the sugars you are trying to create with enzymes. Over-leavened bread isn't ideal, but it isn't the end of the world either. There is a lot to learn about bread baking if you want to really excel at it, but "just bake it, it'll be fine" is always present as you experiment - glad the loaf turned out well!
"No-Knead" bread is a great introduction to "rustic" doughs (very high water content), and to cold fermentation. Cold fermentation is about the best thing you can do for your bread as a home baker, whether or not you are using the no-knead recipe - I cold ferment nearly everything I bake.
If you want to learn more about the process and jump off the deep end, Peter Reinhart's book "The Bread Baker's Apprentice" is a great place to start. It is detailed without being pedantic, simple without being over-simplified, and oriented for the home baker without forgetting the reasons that professionals do things the way they do.
Best Answer
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.