The stuff is coagulated protein, held together by fat. While it won’t hurt you, it won’t taste good and it’s best to remove it on purely aesthetic grounds.
When protein is heated, it coagulates. That is, its long, convoluted molecules unfold and then clump together in new ways. What happened was that some of your chicken’s protein had dissolved in the water where, as the temperature went up, it began to coagulate.
Meanwhile, some of the bird’s fat had melted into oil which, as oil is wont to do, began making its way up to the water’s surface, because it is less dense than water. Wherever the two met, the oil coated the coagulated protein and acted as a life preserver, keeping it afloat as an oily scum. All edible stuff, but not a pretty sight.
As the temperature rises all the way to a simmer, the oil thins out and flows away, leaving the protein to continue clumping. It eventually forms those small brown particles that you can see in the finished soup, that is, if you haven’t removed the scum in its early stages.
The scum hasn’t disappeared; it has just tightened up into those little brown specks, many of which will stick to the sides of the pot at the waterline, forming a kind of (excuse the metaphor) bathtub ring.
So skim away early and diligently and you’ll be rewarded with a nice, clear soup.
The widely recommended slotted spoon for skimming scum from soups and stews isn’t really the best tool, because its holes are too big and it will miss a lot. The best tool for skimming is called (surprise!) a skimmer.
It has a round, flat business end covered with a screen-like mesh. It’s available in kitchenware shops.