Given a method $A(h) = \alpha_0 + \alpha_1 h^k +\mathcal{O}(h^{k+1})$ that approximates $\alpha_0$, the idea is to combine $A$ at varying step sizes to eliminate the lowest power of $h$, say $h^k$.
Since the error of Simpson's rule is $\mathcal{O}(h^4)$, we have $S(h) = \alpha_0 + \alpha_1 h^4 + \mathcal{O}(h^5)$ and \begin{align*} 16 S(h) &= 16 \alpha_0 + 16 \alpha_1 h^4 + \mathcal{O}(h^5) \\\ S(2 h) &= \alpha_0 + 16 \alpha_1 h^4 + \mathcal{O}(h^5) \end{align*} which implies $$16 S(h) - S(2h) = 15\alpha_0 + \mathcal{O}(h^5).$$