[betterunix2]

It means that all polynomials with coefficients in the real numbers have roots in the complex numbers.

Maybe a different example would help. Imagine we are working with Integers modulo 3 (denoted Z/3Z). Z/3Z is a field, meaning we have all the expected arithmetic (+, -, , /) and we can look at polynomials. Here is a polynomial that is irreducible i.e. does not have roots in Z/3Z:

x^2+1

We can construct an extension field in which this has a root; let's call the root i and write the extension Z/3Z(i). Here is where finite fields like Z/3Z differ from the real numbers: extensions like Z/3Z(i) still have irreducible polynomials. On the other hand, in the real numbers, x^2+1 is irreducible, but if we construct an extension where it is reducible, R(i) i.e. the complex numbers, we get a field where there are no irreducible polynomials (that is the basic idea of the fundamental theorem of algebra).