[Rayhem]

It's by no means rigorous, but the gist of it is that

1. You're integrating the polynomial that best approximates the function over the interval. This is why quadrature is exact for polynomials up to some order: the best approximating polynomial is the polynomial itself.

2. Polynomials are good approximations when the function is smooth. Most useful functions are.

3. Because of the smoothness, the behavior in the middle of the interval is largely affected by the behavior at the edges, so you need more densely sample the edges. Think of it like wiggling a string. You're only allowed to wiggle the end, but that very specifically defines the behavior in the middle.

4. There are lots of polynomial sets - Legendre, Chebyshev, Hermite, etc. They're each useful because they're orthogonal to a special weight function, and Legendre polynomials are kind of the default set because they have the simplest weight function: 1.