The study of the quintic by mathematicians such as Lagrange, Abel and Galois led to profound insights that extend well beyond polynomial equations. For example, Galois' proof of the non-existence of a solution by radicals to the general quintic established a critical connection between field theory and group theory (the fundamental theorem of Galois theory). In fact many people view Galois' proof to be the foundation of group theory.
There is an intimate connection between polynomials, geometry and groups. For instance, the symmetries of a regular tetrahedron are described by the symmetric group $S_4$. You can see from the link that when studying this group, polynomials with Galois group $S_4$ can be useful in better understanding the group and/or geometry.
These developments were very influential in the late 1800's, once people understood the significance of what Galois had done (tragically, many years after his death). Sophus Lie noted that his development of Lie theory was motivated by a desire to do for differential equations what Galois had done for algebraic equations.
To your question on motivation, of course none of the major figures leading up to Galois' proof would have fully understood the implications of studying the quintic. But I think Grothendieck's comments (from a personal letter to Faltings) are likely applicable to what was happening,
What my experience of mathematical work has taught me again and again, is that the proof always springs from the insight, and not the other way round – and that the insight itself has its source, first and foremost, in a delicate and obstinate feeling of the relevant entities and concepts and their mutual relations. The guiding thread is the inner coherence of the image which gradually emerges from the mist, as well as its consonance with what is known or foreshadowed from other sources – and it guides all the more surely as the “exigence” of coherence is stronger and more delicate.
At least starting with Laplace and his introduction of resolvents, it seems that the great mathematicians of the time were starting to sense that there was depth to polynomials that needed to be explored.
One final point of clarification, the general quintic is not unsolvable, it is unsolvable by radicals. Hermite discovered that the general quintic can be solved by elliptic modular functions. Klein gives a detailed description of Hermite's solution (and several others), and its connection to the rotation group of the icosahedron ($A_5$), in Lectures on the Ikosahedron, and the Solution of Equations of the Fifth Degree.
