If you are interested in representation theory of finite dimensional algebras (including group algebras and their blocks—and everyone is interested in representations of groups, even if they don't know it), then considering quivers (and bound quivers) is a natural thing to do: all algebras (up to the appropriate equivalence relation relevant in the context of representation theory) are quotients of paths algebras. An immense number of non-finite dimensional algebras are also quotients of path algebras, too.
So whatever motivation you have to be interested in representations of finite dimensional algebras immediately carries over to quivers and their algebras, plus the obvious plus that it becomes extraordinarily easy to build up examples.
Later. As to concrete motivations:
A classical example is the Kronecker-Wierstrass classification of the indecomposable representations of the quiver $\bullet\rightrightarrows\bullet$ , motivated by the conjugation classification of certain systems of ODEs. There is another example, from control theory, in Gabriel-Roiter's book. I am pretty sure these two examples are real, in that the non-representation-theoretic problem came before the representationists took over.
Similarly, the whole cluster explosion of the last 10 years should be a nice example of representation theory of quivers and friends and the methods involved in it helping understand (and solve, in many cases) problems exterior to the theory. Of course, here the source of the problems is also of representation-theoretic nature ---Lie theory--- but in a rather palpable sense this is a quite different part of the theory.
