$\mathscr{O}_X(X-Y)=\mathbb{K}[X]$ if and only if $\text{codim}_X(Y)\geq 2$

Question

I'm having an hard time in solving exercise 3.12 of Gathmann's algebraic geometry book:

Let $X$ be an affine variety such that $\mathbb{K}[X]$ is an UFD and let $Y\subseteq X$ be a non-empty irreducible subvariety. Then: $$\mathscr{O}_X(X-Y)=\mathbb{K}[X]\iff \text{codim}_X(Y)\geq 2.$$

My attempt Given a regular function $f$ on $X-Y$. I'd really like to say that given a point $y\in Y$ then there is a principal open subset $D(h)\subseteq X$ such that $f=\frac{f_\alpha}{g_\alpha}$ in $D(h)-Y$. If $g_\alpha(y)\neq 0$ then we are done. Viceversa, I don't know how to proceed.

EDIT: the answer to the duplicate question uses a scheme-theoretic language and I'm not familiar with it. I want an answer without scheme machinery...