I'm studying "Foundations of algebraic geometry" by Ravi Vakil. Chapter 14.1 is about invertible sheaves on $\mathbb{P}^1_k$. It says between two affine subsets $\operatorname{Spec}k[x_{1/0}]=U_0$ and $\operatorname{Spec}k[x_{0/1}]=U_1 $ there must be transition functions. I don't completely understand if it means transition functions as vector bundles or it's something else;
So far what I think I have understood is that for a locally free sheaf to be invertible, it must be covered with affine open subsets that are of rank 1; Which means here we have $k[x_{0/1}]$ and $k[x_{1/0}]$ as sections of $U_1$ and $U_0$. Their intersection is $k[x_{0/1},x_{1/0}]$ considering $x_{0/1}\sim \dfrac{1}{x_{1/0}}$. And that from a vector bundle and its sheaf of sections, a locally free sheaf can be obtained.
Sorry if it's a very basic question I couldn't figure it out.
$\newcommand{\F}{\mathscr{F}}$$\newcommand{\O}{\mathcal{O}}$The terminology does come from vector bundles but it is more general; transition functions can be thought of as gluing data (cf. Hartshorne exercise II.1.22). In the case of locally free sheaves on a ringed space, we're essentially realizing the sheaf as being a bunch of free sheaves glued together.
Let $\F$ be locally free of rank $n$ on $(X, \O_X)$, where $\{U_i\}$ is a trivializing open cover of $X$. That is, we have an isomorphism $\varphi_i: \O_{U_i}^{\oplus n} \to \F|_{U_i}$ for each $i$. Now, given indices $i, j$ we have composite isomorphisms $$\psi_{ij}: \O_{U_i \cap U_j}^{\oplus n} \stackrel{\phi_i|_{U_i \cap U_j}}\to \F|_{U_i \cap U_j} \stackrel{\phi_j|_{U_i \cap U_j}^{-1}}\to \O_{U_i \cap U_j}^{\oplus n}.$$ These $\{\psi_{ij}\}$ are called the transition functions of $\mathscr{F}$. If you're bored, you can check that the collection $\{\O_{U_i}^{\oplus n}, \psi_{ij}\}$ satisfies the gluing criteria in the quoted exercise, and they glue to form $\F$.