I am reading chapter 11 (on vector bundles) of algebraic geometry I by Wedhorn/Görtz. There they have the following proposition 11.1:
I understand what they write in equation 11.2.1. However, I do not understand the proof of proposition 11.1.
1) How can I prove explicitly that $F$ is a Zariski sheaf?
2) Why does it follow from this that we may assume that $X$ is affine?
3) Then I also don't understand how in this case, we can fall back on equation 11.2.1. In particular, how do I see that $B = \Gamma(X, \mathcal{B})$?
1) First, presheafiness. Let $\alpha : T\to S$ be a morphism of $X$-schemes, with $f_T,f_S$ the structure morphisms. Then $\alpha^{\#} : \newcommand\calO{\mathcal{O}}\calO_S\to \alpha_*\calO_T$ induces a morphism $f_{S,*}\calO_S\to f_{S,*}\alpha_*\calO_T=f_{T,*}\calO_T$, and applying $\newcommand\Hom{\operatorname{Hom}}\Hom_{\calO_X}(\mathscr{B},-)$ gives the presheaf morphism. Explicitly, given $\phi:\mathscr{B}\to f_{S,*}\calO_S$, and $b\in \mathscr{B}(V)$, the homomorphism $\phi' : \mathscr{B}\to f_{T,*}\calO_T$ is defined on $b$ by taking $\alpha^{\#}_{f_S^{-1}(V)}\phi(b)$, which is a section of $\calO_T(f_T^{-1}(V))$, as required.
Let $U_i\subseteq T$ be a covering family for $f:T\to X$. Suppose $\phi : \newcommand\B{\mathscr{B}}\B\to f_*\calO_T$ is a morphism. Then if $V\subseteq X$ is open, and $b\in\mathscr{B}(V)$, $\phi(V)$ is a section of $\calO_T$ on $f^{-1}(V)$, which is determined by its values when restricted to each of the $f^{-1}(V)\cap U_i$, and conversely, given morphisms $\phi_i : \B\to f_*\calO_{U_i}$ (being slightly careless about the fact that I mean the composite of $f$ with the inclusion of $U_i$) that agree on the overlaps, for $b\in\B(V)$, the sections $\phi_i(b)$ agree on overlaps, and thus glue uniquely to give a section $\phi(b)$ of $\calO_T(f^{-1}(V))$.
2) It cites a theorem. Since I don't have a copy of the book in question, I can't explain it, but you might want to check the theorem in the book, and include that in your question if checking the theorem doesn't help you.
However, the rough answer is if a Zariski sheaf is representable when you pullback to open affines, you can glue the representing objects to give a representing object for $X$. This is a sort of descent principle.
3) Notice that in the affine case, morphisms of $\calO_X$-algebras are equivalent to morphisms between their global sections, in which case 11.2.2 reduces to 11.2.1, with $\newcommand\Spec{\operatorname{Spec}}\Spec B$ playing the role of $\Spec \B$. In other words, in the affine case, 11.2.1 says that the object we are looking for is $\Spec B$.