What is the hyperplane bundle (as defined in Fulton's *Young Tableaux*)?

What is the hyperplane bundle (as defined in Fulton's *Young Tableaux*)?

On page 142 of Fulton's Young tableaux, he says that on any projective
space $\Bbb{P}^\ast(V)$ there is a hyperplane line bundle $\mathcal{O}(1)$
described by a quotient line $ V \to L$ whose fiber over a point is the
line $L$. What exactly does this mean?
From what I understand of the hyperplane bundle, it is the bundle over
$\Bbb{P}^n$ whose the total space is constructed from specifying the
transition functions
$$\begin{eqnarray*} \varphi_{ij} : &U_i \cap U_j& \to \Bbb{A}^1 \\
&(x_0,\ldots,x_n)& \mapsto \frac{x_i}{x_j}\end{eqnarray*}$$
where $U_i,U_j$ are the standard affine covers of $\Bbb{P}^n$. More
specifically, the total space $E$ is the disjoint union $\bigsqcup U_i
\times \Bbb{A}^1$ modulo the equivalence relation $(j,u,a) \sim
(i,u,\varphi_{ij}(u)(a))$. How does this construction agree with what
Fulton is saying (and what is he really saying)?