I don't understand the proof of this result in The Red Book of Varieties and Schemes by Mumford.
This result is a corollary to the result which says that every closed immersion of an affine scheme $\textrm{Spec}(A)$ is affine and corresponds to the projection of $A$ onto a quotient $A/\mathfrak a$ for a unique ideal $\mathfrak a$ of $A$.
"Prescheme" means "scheme" in the modern notation. If you can't follow the notation, the condition $\ast$ is saying that for all $y \in X$, there is a neighborhood $U$ of $y$ and elements $s_i \in \mathcal Q(U)$ such that the elements $(s_i)_x \in \mathcal O_x$ generate the ideal $\mathcal Q_x$.
We have a ring homomorphism $R = \mathcal O_X(U) \rightarrow \mathcal O_Y(U \cap Y)$ with kernel $\mathcal Q(U)$. Of course we can enlarge the set $\{s_i\}$ so that these elements generate the entire ideal $\mathcal Q(U)$. I don't understand how we can immediately conclude that $(U \cap Y, \mathcal O_{Y|U \cap Y})$ is affine and isomorphic to $\textrm{Spec}(R/\mathcal Q(U))$.