This is part of Hartshorne's Exercise I.3.6.
I'm having a bit trouble getting started: I see that $k[x, y]$ embeds in $\mathcal{O}(X)$. Now to show the other direction, I'm very confused by what I should do. I read some other solutions online but they don't make that much sense to me (for example, here, I'm confused by why $h$ not being a unit will fail to make $f \in \mathcal{O}(X)$ and there are lots of details unclear in the proof too).
I'm a beginner to Algebraic Geometry so any help would be very appeciated!
The coordinate ring embeds in $k(x,y)$. I'll assume $k$ is algebraically closed. Let $$\frac{f(x,y)}{g(x,y)}\in\mathcal O(X)$$ where $f$ and $g$ are polynomials. We may assume (since $k[x,y]$ is a UFD) that $f$ and $g$ have no common factors. Suppose that $g$ is not constant. Then $g(x,y)=0$ has infinitely many solutions in $\mathbb A^2(k)$. By Bezout's theorem, $f$ and $g$ have only finitely many common solutions. So $g(x,y)=0$ has a solution $(x_0,y_0)$ which is not $(0,0)$ nor is it a solution of $f(x,y)=0$, so $f(x_0,y_0)\ne0$. So $f/g$ is not defined at $(x_0,y_0)\in X$, and so $f/g\notin\mathcal O(X)$.