I am trying to understand the Lemma 29.3.1.
We begin with a quasi-compact scheme. Then we have the following construction:
Let $X$ be a scheme. $x \in X$ a closed point, $U=Spec(A)\subseteq X$ an affine neighborhood. $Z=X \setminus U, Z'=X \cup \{x \}$. There are quasi-coherent sheaves of ideals $I$, $I'$ cutting out $Z$ and $Z'$. Giving SES, $$ 0 \rightarrow I' \rightarrow I \rightarrow I/I' \rightarrow 0 $$
In line 9 of proof it is claimed that
$I/I'$ on $U$ corresponds to the $A$-module $A/m$, where $m$ is the maximal ideal in $A$ corresponding to $x$.
I don't understand why.
We know as presheaves, $I(U)=A, I'(U)=m$. But $I/I'$ is the sheafiffication(?), how do we know $I/I'(U)=A/m$?
The short answer is that on an affine scheme, everything about a quasicoherent sheaf is determined by its global sections. Indeed, take the SES of sheaves mentioned in your post and take sections over $U$. Since $U$ is affine, all higher cohomologies vanish, so one gets that $0\to I(U)\to I'(U) \to (I/I')(U) \to 0$ is exact, which shows via the 5-lemma that $(I/I')(U)\cong I(U)/I'(U)$. But this final term is just $A/\mathfrak{m}$.