Let X and Y be integral k-scheme of finite type.Let $x \in X$ and $y \in Y$ and let $\phi:\mathcal{O}_{Y,y} \rightarrow \mathcal{O}_{X,x}$ be an isomorphism of k-algebras. Then, Claim: There exists an open neighborhood U of x in X and V of y in Y and an isomorphism $h: U \rightarrow V$ of k-schemes with h(x)=y such that $h_x^{\#}=\phi$
Proof: Assume X=Spec B and Y=Spec A. Let $\rho \subset A$ and $q \subset B$ be the prime ideals corresponding to y and x respectively. Then, we have an isomorphism $\phi: A_{\rho} \rightarrow B_{q}$. This induces isomorphism $Frac(A) \rightarrow Frac(B)$. Till here I am fine but I am not sure what to do after this.
$\textbf{I don't understand why the next praragraph is true ?}\\$
Since A and B are finitely generated, we can find elements $f \in A $ and $g' \in B$ such that $\phi(A) \subseteq B_{g'} \subseteq \phi(A_f)$, and since $\phi(A) \subset \phi(A_{\rho})=B_q \supset B$, we can choose $f \in A \backslash \rho, g' \in B \backslash q$. But then for suitable $n > 0,g:=(g')^n \phi(f)$ lies in $B \backslash q$, too, and $\phi(A_f)=B_g$. Thus $\phi$ yields an isomorphism $h:U:=D(g) \rightarrow V:=D(f)$
