No zeros or poles implies regular

Question

The following is Theorem 8.14 of Milne's notes (page 176): Let $V$ be a normal variety, let $ f $ be a rational function on $V$. If $f$ has no zeros or poles on an open subset $U$ of $V$, then $f$ is regular on $U$. The proof says to apply a commutative algebra result: a noetherian domain $A$ is normal if and only if $A_p$ is a DVR for all prime ideals $p$ of height $1$ and $A = \bigcap_{\text{ht}(p) = 1} A_p$. I don't see how this implies the theorem or how it is related to the prior discussion of divisors and Picard group. I hope someone can help me clarify this exposition.