This is related to Ueno's Algebraic Geometry 2, Chpt 5, Sec 3, (b) Segre morphism.
$X= \operatorname{Proj}(k[x_0,x_1])$. Let $f:X\to \operatorname{Spec}(k)$ be structural morphism. Let $V_1=\Gamma(X,O_X(1))=f_\star(O_X(1))$.
$\textbf{Q:}$ Why $f^\star(V_1)=O_{X,x_0}\oplus O_{X,x_1}$? $V_1$ is really 2 dimensional vector space. So $V_1=k^2$. Now $f^\star(V_1)=f^\star(k\oplus k)=f^\star(k)\oplus f^\star(k)$. It suffices to consider $f^\star(k)$. $f^\star(k)=f^{-1}(k)\otimes_{f^{-1}(k)}O_X$. Restrict to affine level, it is obvious that $f^\star(k)|_{x_0\neq 0}=k[\frac{x_1}{x_0}]$. Then glue to give $O_X$. What is the meaning of notation $O_{X,x_0}$? $x_0=0$ or $x_0\neq 0$? My guess is that this is local ring. The other interpretation is that $x_0$ is a prime ideal.