The well is used for methods where you work without exact proportions from a recipe.
You start by mixing everything except the flour into your wet ingredients. Yeast, salt, eggs, milk, water, sugar - whatever your recipe needs, it is quirled together. Then you take a large amount of flour and place it either on the counter as Sneftel said, or in a bowl. Or you use the vessel in which you hold your flour anyway (which may be some kind of small vat holding 5-10 kg of flour).
Here comes the well. You make the well in your piled-up flour, it looks a bit like the crater in a volcano. Then you add a little bit of your liquid and mix, scooping just enough flour from the bottom of the well to get a liquid but viscous dough. You continue adding liquid from above and extending your well in the depth and width until all your liquid is added. From then on, you continue the mixing process with gradual addition of flour until you can start making kneading motions. Then you knead the soft dough within its well. Once it has reached the correct consistency, you can take it out of the well and knead further on a floured surface. The well stays where it is, you end up not using most of the flour you started with.
The advantages of this method are:
- you don't have to measure your flour or even transmit the information on how much flour you need through a recipe - you rely on your knowledge of baking to decide when the dough is right *
- you have a significantly reduced risk of lumps when compared to dumping everything at once in a bowl and starting to stir **
- you indeed don't get any sticking to the bowl, since the dough never touches the sides of the bowl - it only touches the sides of the well
- if you are using your counter or your flour vat, you don't have to move your flour around or to wash a bowl
- it is popular ***
- For completeness, I'll repeat the advantage first brought up by Sneftel: It allows you to make bread on a counter or table, without the need to have a bowl
* This is a disadvantage from the perspective of a person with high literacy, easy access to standardized measuring tools, and not-so-advanced baking skills. It was an advantage for the people who established the method, though.
** This point is obsolete if you are using an electric mixer
*** Yes, I consider being popular to be an advantage in its own right, beyond the technical reasons which might have lead to the popularity in the first place. Discussion on this is not culinary-specific and goes beyond the scope of the answer or even the site.