consists entirely of isometries of the hyperbolic plane.
I'm not even sure what this means, and all I can find is the same statement everywhere. Can somebody provide a proof or link to one?
If I'm not mistaken, it is rather .
Any holomorphic bijection of the upper half-plane to itself is a Möbius transformation with real coefficients. Essentially, you can extend such a map, using the Schwarz reflection principle, to a conformal mapping of the sphere to itself fixing the real line. It's because of this that all coefficients must be real.
Now to show that these are in fact isometries and not just homeomorphisms, you need to look at how the distance is defined. It essentially depends only on the cross-ratio between the two points and the endpoints "at infinity" of the line joining them. Since the cross ratio is invariant under Möbius transformations, it follows that these maps preserve distance.