I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous parallel lines postulate, putting an end to centuries of unsuccesfull attempts to deduce the last axiom from the first ones.
It seems to be, apart from this fact, of genuine interest since it was part of the usual curriculum of all mathematicians at the begining of the century and also because there are so many books on the subject.
However, I have not found mention of applications of hyperbolic geometry to other branches of mathematics in the few books I have sampled. Do you know any or where I could find them?
Maybe this isn't the sort of answer you were looking for, but I find it striking how often hyperbolic geometry shows up in nature. For instance, you can see some characteristically hyperbolic "crinkling" on lettuce leaves and jellyfish tentacles:
My guess as to why this shows up again and again (and I am certainly not a biologist here, so this is only speculation) is that hyperbolic space manages to pack in more surface area within a given radius than flat or positively curved geometries; perhaps this allows lettuce leaves or jellyfish tentacles to absorb nutrients more effectively or something.
EDIT: In response to the OP's comment, I'll say a little bit more about how these relate to hyperbolic geometry.
One way to detect the curvature of your surface is to look at what the surface area of a circle of a given radius is. In flat (Euclidean) space, we all know that the formula is given by $A(r) = \pi r^2$, so that there is a quadratic relationship between the radius of your circle and the area enclosed. Off the top of my head, I don't know what the formula is for a circle inscribed on the sphere (a positively-curved surface) is, but we can get an indication that circles in positive curvature enclose less area than in flat space as follows: the upper hemisphere on a sphere of radius 1 is a spherical circle of radius $\pi/2$, since the distance from the north pole to the equator, walking along the surface of the sphere, is $\pi/2$. In flat space, this circle would enclose an area of $\pi^3/4 \approx 7.75$. But the upper hemisphere has a surface area of $2 \pi \approx 6.28$.
By contrast, in hyperbolic space, a circle of a fixed radius packs in more surface area than its flat or positively-curved counterpart; you can see this explicitly, for example, by putting a hyperbolic metric on the unit disk or the upper half-plane, where you will compute that a hyperbolic circle has area that grows exponentially with the radius.
So what happens when you have a hyperbolic surface sitting inside three-dimensional space? Well, all that extra surface area has to go somewhere, and things naturally "crinkle up". If you are at all interested, you can crochet hyperbolic planes (see, for instance, this article of David Henderson and Daina Taimina), and you'll see how this happens in practice.