I like to think about this geometrically. $H/\Gamma$ is a topological metric space. At most points of $H$ (that are not fixed by any element of $\Gamma$, the quotient looks just like the hyperbolic plane $H$ itself. The singularities come from elliptic elements of $\Gamma$, i.e., (locally) rotations, where you get a cone metric (locally like the metric on a piece of paper rolled into a cone). Remove those points and consider the conformal structure coming from the resulting Riemannian metric. The ends of this surface look like removable singularities (locally like $\mathbf{C}$ minus a point in their conformal structure), and so can be filled in uniquely. $X$ is the result of filling in the points, and $S$ is the set of filled in points.

The points in $S$ come both from the removed cone points and from some points at infinity, the elliptic points as Charlie explains above.

You have to be careful to distinguish between removing a point (leaving a removable singularity) and removing a small ball (which is different conformally). Removing a small ball always gives a surface with infinite hyperbolic area when uniformized, and is never equivalent to a compact surface minus points.