From Hartshorne, Chapter II.2, Proposition 2.5(b).
If $R$ is a graded ring and $\mathfrak a$ is a homogenous ideal, then the function defined as
$$\phi(\mathfrak a) = \mathfrak aR_f\cap R_{(f)}$$
is a bijection, where $R_f$ is the usual localization of $R$ with respect to the multiplicative set $S=\{1,f,f²,...\}$ and $R_{(f)}$ is the subring of homogeneous elements of degree $0$ of $R_f$.
In EGA II, Proposition 2.3.6, there is a nice proof of the fact that the map $\psi_f: D_+(f)\rightarrow\operatorname{Spec}(S_{(f)})$ given by $\mathfrak{p}\rightarrow \mathfrak{p}_f\cap S_{(f)}$ is a homeomorphism.
Let $f\in S_d$. The map $\psi_f$ is continuous because for $g\in S_e$ you have $g^d/f^e\notin\psi_f(\mathfrak{p})=\{x/f^m:x\in \mathfrak{p}_{md}\}$ if and only if $g\notin\mathfrak{p}$, and the sets $D(g^d/f^e)$ for $g\in S_e$, $e>0$, form a basis for $\operatorname{Spec}(S_{(f)})$ (for any $x/f^m\in S_{(f)}$, $D(x/f^m)=D(x^d/f^{md})=D(x^{\deg(f)}/f^{\deg(x)})$). Once we proved that $\psi_f$ is injective, then this will imply that $\psi_f$ is a homeomorphism of $D_+(f)$ with its image (the sets $D_+(fg)$ form a basis for the topology of $D_+(f)$).
To show that $\psi_f$ is injective, since $D_+(f)$ is $T_0$ (it has the induced topology of $\operatorname{Spec}(S)$), given two elements in $D_+(f)$ there exists $g$ , say of degree $e$, such that $D_+(fg)$ contains one of them but not the other. According to what we have proved above, just the image of one of them is contained in $D(g^d/f^e)$.
Finally, to show that $\psi_f$ is surjective, given $\mathfrak{q}_0\in\operatorname{Spec}(S_{(f)})$, let $\mathfrak{p}_n=\{x\in S_n:x^d/f^n\in\mathfrak{q}_0\}$ and check that $\mathfrak{p}=\oplus\mathfrak{p}_n$ is a prime ideal in $D_+(f)$ such that $\psi(\mathfrak{p})=\mathfrak{q}_0$.
