Let $\pi:\mathbb{C}^{n+1}\smallsetminus 0 \rightarrow \mathbb{P}^n$ be the canonical projection, let $\mathcal{O}(1)$ be the hyperplane bundle over $\mathbb{P}^n$ and let $(x_0,\dots,x_n)$ be the canonical coordinates on $\mathbb{C}^{n+1}$. For $n\geq 0$ denote $\mathcal{O}(n)=\mathcal{O}(1)^{\otimes(n)}$
Suppose $V=\sum_{i=0}^n P_i \frac{\partial}{\partial x_i}$ is a vector field on $\mathbb{C}^{n+1}$ where each $P_i$ is a homogeneous polynomial in $x_0,\dots, x_n$ of degree $r$. Then, in why can $$\pi_*(V)=\sum_{i=0}^n P_i \pi_\star \big(\frac{\partial}{\partial x_i}\big)$$ be interpreted as a global section of the bundle $ \mathcal{O}(r-1) \otimes T \mathbb{P}^n$? Or (equivalently, I hope), why is $\pi_*(V)$ the product of a homogeneous polynomial of degree $r-1$ with a global vector field on $\mathbb{P}^n$?
The reason for this question is that I'm trying to understand the following extract:
Let $\Theta_{\mathbb{P}^n}$ and $\mathcal{H}$ be the tangent and hyperplanes sheaves on $\mathbb{P}^n$. For an $\mathcal{O} _{\mathbb{P}^n}$-sheave $\mathcal E$ let $\mathcal E(d) = \mathcal{E} \otimes \mathcal{H}^{\otimes d}$ if $d\geq 0$ and and $\mathcal{E} \otimes \mathcal (\mathcal{H} ^*)^{-d}$ if $d\leq 0$. Let $\Pi:\mathbb{C}^{n+1}\smallsetminus 0 \rightarrow \mathbb{P}^n$ be the natural projection. The twisted Euler sequence is then
$$ 0 \rightarrow \mathcal{O}_{\mathbb{P}^n} (r-1) \stackrel{\phi}{\longrightarrow} \mathcal{O}_{\mathbb{P}^n}(r)^{\oplus (n+1)} \stackrel{\Pi_*}{\longrightarrow} \Theta_{\mathbb{P}^n} (r-1) \rightarrow 0 . $$
But the codomain for $\Pi_*$ doesn't make sense to me.
I'm very new to this topic and have been stumbling hard, I'd appreciate any help.