I'm reading Griffith & Harris book, there is an identification which I confused:
Let $V$ be a complex vector Space of dimension $n$, consider its projectivization $\mathbb{P}(V)=V^\times/\mathbb{C}^*\cong\mathbb{P}^{n-1}$, then the global sections of the hyperplane bundle can be identified with the dual Space: $$H^0(\mathbb{P}(V), \mathcal{O}(1))=V^*$$
I don't know how to identify these two spaces, I can just see that they are of same dimension.
One can identify global sections of $O(n)$ with homogeneous polynomials of degree $n$ in a choice of basis of $V$, or abstractly, as the degree $n$ polynomial functions on $V$. So if one is comfortable with this, taking $n=1$ gives that the space of sections is naturally just linear polynomial functions on $V$, which is just $V^*$. It remains to see why we ought to have this description, and there are a few ways, depending on your background. One way is algebra-geometrically, via the construction of projective space using the Proj construction, then the associated graded module corresponding to $O(n)$ is the nth shift down of the graded ring input, in our case polynomial functions on $V$, graded by degree. Another more direct way to see this is to use transition functions, we can write down the cocycles explicitly for this line bundle, and doing it in coordinates gives polynomials of the correct degree. This is best done with the dimension equal to $2$ to see the idea, it’s a fantastic exercise for using charts.
Any textbook on algebraic geometry covers at least one of these approaches, usually both, I know Hartshorne does.