This is a passage from a complex geometry lecture notes
Any homogeneous polynomial $s \in \mathbb{C}[z_0,...,z_n]_k$ with degree $k$ defines a linear map $(\mathbb{C}^{n+1})^{\otimes k} \to \mathbb{C}$. This gives rise to a holomorphic map $\mathbb{P}^n \times (\mathbb{C}^{n+1})^{\otimes k} \to \mathbb{C}$ which is linear on any fibre of the projection to $\mathbb{P}^n$. The restriction to $\mathcal{O}(-k)$ thus provides a holomorphic section of $\mathcal{O}(k)$.
I need some help understanding this result. Why does a homogeneous polynomial $s \in \mathbb{C}[z_0,...,z_n]_k$ with degree $k$ give us a linear map $(\mathbb{C}^{n+1})^{\otimes k} \to \mathbb{C}$ and secondly why does this give rise to a holomorphic map $\mathbb{P}^n \times (\mathbb{C}^{n+1})^{\otimes k} \to \mathbb{C}$?
It seems that the author is skipping over some important details here.