Polynomials turn the idea of performing arithmetic operations (a dynamic, procedural notion: performing operations) into a static object of mathematics that can be considered in its own right (a static notion: a list of instructions, or an element of a ring). This can be seen as an early example of reifiction/object-ification in mathematics, which has traditionally been quite useful, viz. the definition of algorithm, the definition of ring, the definition of category and functor, the definition of cardinal+ordinal, etc. These all take what was previously a rough or vague or intuitive or unstated or procedural/dynamic mathematical idea and turn it into a single mathematical object that can then be studied as such. That is, one can study the object in its own right, its relation with other objects of the same or different types, etc. (In particular, turning processes into objects has been quite useful, but this is but one kind of a more general phenomenon.)

Now, why are polynomials specifically so incredibly and widely useful? Perhaps because the thing they are reifying is so basic, namely the arithmetic operations.