How do even and odd functions relate to even and odd numbers?

They share lots in common!

1. The product of two even functions is even, the product of an even function and an odd function is odd, and the product of an odd and an even function is odd. This is exactly how even and odd numbers add, and is a nice motivating example of an isomorphism.

2. The Taylor series (at $x = 0$) of an even function has only even terms $x^{2n}$, and the Taylor series of an odd function has only odd terms $x^{2n+1}$. If you don't know what a Taylor series is, we can restrict our attention to polynomials: a polynomial is even iff it has only even terms, and a polynomial is odd iff it has only odd terms.

For an even function $f(-x)=f(x).$ Now you know that $(-x)^n=x^n$ if $n$ is even.

For an odd function $g(-x)=-g(x)$. Now you also know that $(-x)^n=-x^n$ if $n$ is odd.