Radian measure is important to mathematics, especially trigonometry and calculus, in the sense that it provides a "pure," or unitless, measurement of an angle. This allows for very simple expression of derivative and integral relations that involve trigonometric functions in calculus. In trigonometry radian measures help highlight many beautiful symmetries between certain types of angles and triangles, and the trigonometric functions. As radians relate angles, circles (more precisely, arc length) and triangles, they also provide a coherent foundation for geometry.
Most basically, radian measure is used to quantify the relation between a circle and an inscribed angle, and as such is a useful function relating angles and lengths. Specifically, the radian measure of an angle is the ratio of length of the arc contained by the angle to the radius of the circle, making a full circle 2*pi (radians). Radian measure is used throughout pure mathematics, especially in situations where being able to manipulate variables without consideration for units is beneficial.
What makes radian measure distinct from most types of measurement systems is that describes a ratio of something relative to itself, instead of to some base measure, for example, a degree, or a certain unit of length. While this may sound slightly counterintuitive, it is this very property that frees radians from unit dependency and makes them attractive for applications within mathematics. They also naturally arise from a variety of functions in calculus, simply because they are unitless; for example, in describing the motion of a point on a wheel.
Radian measure clarifies a great deal of trigonometric relations and simplifies a number of relations in calculus. This means one can eliminate many constants that have to be added in otherwise in trigonometric and other geometric relationships, which reduces the number of calculations that have to be done (perhaps less beneficial in the age of calculators and computers, but at least fewer numbers to keep track of). In calculus, having fewer types of units to keep track of is indispensable, and indeed in many cases the computations cannot be done with degrees at all -- they must be converted to radians and then converted back if a degree result is required.
While radian measure may appear to be the most intuitive method for describing angles from a geometric standpoint, the numbers that it works with are less than intuitive to most people, especially for angles that do not fall into a few privileged categories, nor are they easy to work with in more practical settings such as actually measuring angles. As such, radian measure is generally not used outside of pure mathematics, degree measure being the preferred system in most of the natural sciences and engineering disciplines.