Let $S$ be a finitely generated graded $A$-algebra. For each homogeneous $f\in S_+$, we have a scheme structure $D(f)\cong \operatorname{Spec} S_{(f)}$ where $S_{(f)}$ denotes the zeroth piece of the graded localization ring $S_f$. These glue well and give rise to a scheme structure on $\operatorname{S}$.
Now my question is how to understand the ring $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$? In particular, if $S = k[x_1,\cdots, x_n]$, then what is the ring $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$?
To compute this ring, use the sheaf axiom -- global sections are just sections on an open cover that agree on the overlap.
As an exercise, you might try to prove in your specific example what you already know from the classical formulation, namely, that the global sections of the structure sheaf on projective space over a field is just $k$. (The only global regular functions on projective space are constant.)