When solving high-school algebra problems, we try to reduce them to arithmetic. When solving calculus problems, we try to reduce them to algebra. When solving differential equations, we try to reduce them to integrals. In essence, we use some new technique in order to take a problem we don't know how to solve and transform it into a problem we do know how to solve.

That's why we factor: because we can (sometimes) transform a higher-order equation into a series of linear pieces, and solving a linear equation is essentially arithmetic.