You can do many more things with measurable functions that with nonmeasurable ones, so the theory is richer and has more applications. There's much less trouble with integrals, for example, right? In fact, if in the beginning the measurability seems a necessary evil, in the end it becomes a blessing and helps interpreting things. E.g., if a random variable $Y$ is measurable with respect to a sigma-algebra generated by a random variable $X$ then $Y$ can be expressed in terms of $X$, i.e., $Y=f(X)$ for some $f$. Similarly, many concepts of dependence like the Markov property, martingale property, notions from stochastic control, etc. are natural to formulate in terms of measurability w.r.t. appropriately chosen sigma-algebras.

You can do many more things with measurable functions than with nonmeasurable ones, so the theory is richer and has more applications. There's much less trouble with integrals, for example, right? In fact, if in the beginning the measurability seems a necessary evil, in the end it becomes a blessing and helps interpreting things. E.g., if a random variable $Y$ is measurable with respect to a sigma-algebra generated by a random variable $X$ then $Y$ can be expressed in terms of $X$, i.e., $Y=f(X)$ for some $f$. Similarly, many concepts of dependence like the Markov property, martingale property, notions from stochastic control, etc. are natural to formulate in terms of measurability w.r.t. appropriately chosen sigma-algebras.