In the first application, we have a picture (embedded in the complex plane) which is the following situation:
Each of the yellow points represents where the circles touch each other. The statement of this theorem is that, as we continue this process indefinitely, all of the yellow points will lie on a circle. To see this, we choose a Möbius transformation which ==1;;sends the point where both the large circles touch to $\infty$==. Then the two big circles become lines, and all of the smaller circles are inbetween these two parallel lines, so they all have the same size.This next application goes by the name of Steiner's Porism. I suppose Porisms are generally theorems of the flavor of "If it holds for one case, it holds for all cases". In this case we have the following situation:
Now we want to fill in the between area with circles, similary to before. But this time, since the circumference is bounded below, the number of circles must terminate. The question is, could we have picked a different order to fill in the circles so that they close in the end? Steiner's Porism says that if there is one way to fill in the circles so that they circles touch/don't touch, then there isn't any way to fill in the circles so that they touch. The trick here is to apply a Möbius transformation which makes the two circles concentric, and then all of the intermediate circles would need to be the same size.