At risk of being overly bold, allow me to suggest:
Polynomials are useful because quadratic polynomials are useful.
If we can all agree that linear algebra is an indispensable tool in mathematics then it's hard to argue with the success of equipping vector spaces with quadratic structures - this is the starting point of nearly all of geometry and large portions of number theory. Even when we move on to higher degree polynomials or transcendental objects, quadratic structures generally appear as local approximations (how often do you go past the degree 2 term in a Taylor series?)
So the question becomes: why are quadratic polynomials useful? There seem to be two different but interacting reasons. The first is that quadratic functions of a real variable are always either convex or concave and therefore have a unique maximum or minimum. The second is that quadratic functions are intimately related to bilinear forms and therefore can be accessed using linear algebra. The combination of these two reasons seems to explain the success of quadratic algebra in analysis and geometry (e.g. Hilbert spaces, Riemannian manifolds). This is also part of the story behind their utility in number theory, though I'm not sure I've completely explained the importance of quadratic structures on finite fields. (Related issue: why is quadratic algebra over $\mathbb{F}_2$ so fundamental in the topology of manifolds?)