A good reference is S. Lang, Introduction to complex hyperbolic geometry.
Shortly, I can give the following reasons. Listed in historical order.

There are famous Picard theorems about entire curves whose target is of dimension 1.
They are 130 years old and they have a lot of connections with other areas of mathematics.
How to generalize them to targets of dimension more than 1 ? There are many conjectures but
most of them are still conjectures.

In dimension 1, we have this very useful thing which is called hyperbolic metric. It is one of
the main tools in the theory of analytic functions. There is an analog in higher dimensions,
called Kobayashi metric. It is also very useful, but the question is: which complex manifolds
have it? The answer (for compact manifolds) is given by Brody's Lemma: such metric on a compact
manifold exists if and
only if there are no non-constant holomorphic curves C to X. That is it reduces the question
to the question about holomorphic curves.

There is a deep analogy between the theory of holomorphic curves and Diophantine geometry.
That is (very roughly speaking) a manifold over an algebraic number field has finitely many
rational points if, as a complex manifold, it is hyperbolic. That is it has no non-constant
holomorphic curves.

Nevanlinna theory is one of the main tools of study of holomorphic curves. It was originally
developed in the attemts to better understand the nature of Picard's theorems.

This is a very brief answer, and I recommend the book of Lang mentioned above, and his survey
articles on holomorphic curves and diophantine geometry. For one-dimensional theory, which is
extremelly rich and developed I recommend the books of Nevanlinna himself. They better explain
the original motivations then the modern books.

needcome from? Who told you that youneedto study entire curves? – Qiaochu Yuan Sep 18 '12 at 3:27