We need to show that the map
$$\theta:\mathbb{C}[z_0,\ldots,z_n]_k\to H^0(\mathbb{P}^n,\mathcal{O}(k)) $$
Is an isomorphism.
Huybrecht's describes the map $\theta$ right before the proposition.
This is what I understand from the description:
Given a homogeneous polynomial of degree $k$ we can associate it to a linear map $(\mathbb{C}^{n+1})^{\otimes k}\to \mathbb{C}$, what we call Polarization of an algebraic form. So the map here it's just the evaluation of the "polar form".
Now we can look at the product $ \mathbb{P}^n\times (\mathbb{C}^{n+1})^{\otimes k}$, a natural question here is why ? The answer I give is because we want to end up with a section of $\mathcal{O}(k)$, and there is the inclusion $\mathcal{O}(-k)\subset \mathbb{P}^n\times (\mathbb{C}^{n+1})^{\otimes k}$.
So this gives rise to a holomorphic map $\mathbb{P}^n\times (\mathbb{C}^{n+1})^{\otimes k}\to \mathbb{C}$. What is the map here ? ( I don't know), but I don't think we care too much, all that matters is that we can always go form a vector space to its field of scalars, what we call "linear form", possible thought some kind of evaluation.
Now we come to the hole point. We can also consider the projection of $\pi:\mathbb{P}^n\times (\mathbb{C}^{n+1})^{\otimes k} \to\mathbb{P}^n$. So if you look at the fiber of this projection and restrict it to $\mathcal{O}(-k)$ you end up with a section of $\mathcal{O}(k)$. How ?
From what I understand, we take restriction to $\mathcal{O}(-k)$ of the fiber of $\pi $ and we compose with the natural isomorphism of the dual of the line bundle. So we end up with a map $\mathbb{P}^n\to \mathcal{O}(k)$, a section ?
I don't feel that's what we mean here, how the previous constructions play any role to this? What we need the map $\mathbb{P}^n\times (\mathbb{C}^{n+1})^{\otimes k}\to \mathbb{C}$ for ?
Later in the proof he pulls out of the hat a composition $\mathcal{O}(-1)\subset \mathbb{P}^n\times \mathbb{C}^{n+1}\to \mathbb{C}^{n+1}$ I have no idea what that composition is, all I can think of related to this that the fiber $\mathcal{O}(-1)$ over $l\in \mathbb{P}^n$ is $l\subset \mathbb{C}^{n+1}$.