What happens if a function is not measurable? Definition aside That is, what would be the motivation for mathematicians to restrict their condition to only measurable functions. Desire for integrability is one. What else?
 A: Every branch of mathematics uses certain sets as their basic building blocks. In measure theory these are measurable spaces, in topology topological spaces, in linear algebra vector spaces etc.
That we consider measurable spaces (i.e. a spaces endowed with a $\sigma$-algebra on them, i.e. pairs $(\Omega, \mathcal{A})$) has to do with the basic abstractions of what a volume is, which leads to the notion of a measure. It turns out that the best way to think about which sets to measure, i.e. take "volumes" of, leads to $\sigma$-algebras.
Now we would like to consider maps between several such measurable spaces, i.e. $f \colon (\Omega_{1}, \mathcal{A}_{1}) \rightarrow (\Omega_{2}, \mathcal{A}_{2})$ that are "compatible" with their $\sigma$-algebras. One case of particular importance is that where you have a measure on the domain space, i.e. $f \colon (\Omega_{1}, \mathcal{A}_{1}, \mu_{1}) \rightarrow (\Omega_{2}, \mathcal{A}_{2})$. You would like to use this to define a measure $\mu_{2}$ on your target space. How would you do that? 
The most natural way is to define (note that, by what I wrote above, measures "live" on $\sigma$-algebras, so we need to know what $\mu_{2}$ is on any sets from the $\sigma$-algebra $\mathcal{A}_{2}$)
$$
\mu_{2}(A_{2}) := \mu_{1}(f^{-1}(A_{2})).
$$
But for this to make sense, as $\mu_{1}$ "lives" on the $\sigma$-algebra $\mathcal{A}_{1}$, we need the following condition:
$$
\forall A_{2} \in \mathcal{A}_{2}: \quad f^{-1}(A_{2}) \in \mathcal{A}_{1},
$$
i.e. the measurability of the map $f$.
Edit: What @HennoBrandsma has been nodding towards is the notion of a category. Simply speaking, in category theory you look at categories, which are the classes of objects (typically on the level of sets) that are of interest, and the ones that I mentioned in the beginning of my post (topological spaces etc.) are precisely the objects in these categories. But for things to get interesting, you need to pass from one object to another in a meaningful way. These mappings are called morphisms, and in each category these morphisms have different properties, all of which ensure that the structures on you objects are compatible with these mappings. So for example in the category of vector spaces (which have a linear structure, i.e. you can add and scale vectors), the morphisms are the maps that respect this linear structure, i.e. linear maps or homomorphisms. In topology the maps that respect the topology are continuous maps, and in measure theory your morphisms are measurable maps. This "categorical thinking" provides a nice way to summarise a lot of results and concepts in various branches of mathematics.
So if you have enough experience, you expect that there are maps that preserve the structure of your objects and that these maps play an important role. But this was all abstract nonsense, so if that is too abstract for you, just stick to the explanation given above.
