Methods based on Taylor series using $N$ points will integrate exactly a polynomial $P$ of degree $N-1$. If a function $f(x)$ can be approximated with a polynomial of degree $N-1$ $$\begin{equation*} f(x)\approx P_{N-1}(x), \end{equation*}$$ with $N$ mesh points we should be able to integrate exactly the polynomial $P_{N-1}$.

Gaussian quadrature methods promise more than this. We can get a better polynomial approximation with order greater than $N$ to $f(x)$ and still get away with only $N$ mesh points. More precisely, we approximate $$\begin{equation*} f(x) \approx P_{2N-1}(x), \end{equation*}$$ and with only $N$ mesh points these methods promise that $$\begin{equation*} \int f(x)dx \approx \int P_{2N-1}(x)dx=\sum_{i=0}^{N-1} P_{2N-1}(x_i)\omega_i, \end{equation*}$$