Invariant metrics and pseudometrics are used to study holomorphic maps between complex manifolds. The model is the use of the hyperbolic metric in dimension 1.
For example, a manifold is called hyperbolic if its Kobayashi pseudometric is a true metric. Then every holomorphic map from a line to a hyperbolic manifold is constant. This is a generalization of Picard's Little Theorem. Remarkably, for compact complex manifolds there is also the converse statement (Brody'a Lemma).
There are also generalizations of Picard's Great Theorem, normality criteria and generalizations of removable singularity theorem. One major application is to the proof that the group of automorphisms of a compact hyperbolic space is finite.
All this has a lot of further applications, though much less than for the hyperbolic metric in one-dimensional case.
I recommend an excellent survey by Kobayashi himself, as well as his books:
Hyperbolic manifolds and holomorphic mappings, 1970 and Hyperbolic complex spaces, 1998, and the book by S. Lang, Introduction to complex hyperbolic spaces.