The trapezoid rule works by approximating any given function, over small enough intervals, as lines, and then adding up the areas of the resulting trapezoids. The idea behind Simpson's rule is the exact same: just as we can approximate over a small interval as a line, so can we approximate as a parabola. As long as we know the formula for the area under a section of any parabola (which has been well-known since antiquity), we can approximate the total area under the curve by slicing it up and adding up areas of small parabolas. Lines are curves too, but they need two points to "fix" them; quadratics need three points, but otherwise they're just like lines. Just like adding up areas of trapezoids, we can also add up areas of parabolas, and it works even more accurately.