Why am I learning model theory?

That's a really interesting question on which we could talk for days, but I hope that in this answer I could tell enough stuff to convince you of the importance of model theory.

The great importance of model theory come from the facts that lots of mathematical structures can be described as models of a first-order theory. One of the more useful things model theory do to us is to translate property of structures into properties of theories (i.e. sets of formulas), which are usually easier to prove to the discrete nature of theories. Using model theory we can prove a lot of things about mathematical structures in a very easy way considering structures as models of some theory.

Here's an example: every graph $G$, even an infinite one, can $k$-colored iff every finite sub-graph can be $k$-colered. This theorem is clearly trivial for finite graphs, nonetheless the only proof I know of this fact uses the compactness theorem.

Lot of algebraic geometry (if not all) over an algebraically closed field of characteristic $0$ can be reduced to the study of algebraic geometry over $\mathbb C$. That's because $\mathbb C$ is a $\mathfrak c$-categorical model of the theory of algebraically closed field of characteristic $0$, and so every first-order property is true for every algebraically closed field of characteristic $0$ iff is true for $\mathbb C$.

Model theory give a lot of techniques for build up models. This is extremely helpful to prove existence of structures of given kind, for instance algebraic ones, that can be described as model of first order theory. On the other end this techniques can also be used in order to prove that some result don't generally hold, by creating counterexamples.

As I said in the beginning there are more stuff could be said about this but I hope this is sufficient to explain way model theory is really important, especially for application in mathematics.

One advantage is that model theory clarifies the informal distinction between syntactic and semantic aspects of modern mathematics, making it possible to give precise solutions to problems that remained intractable before the advent of model theory. My favorite one is Abraham Robinson's solution of the 300-year-old problem of the consistency of infinitesimals. In the framework of model theory one can give precise meaning to terms such as non-standard models of analysis and successfully formalize centuries of work by the pioneers of analysis who thought of the derivative as a ratio of infinitesimals and of the integral as an infinite sum.

For a discussion of examples of infinitesimals see this question and the answers there.