I wonder why every birational transformation of $\mathbb{P}^n_k$ can be written as $[f_1:...:f_{n+1}]$, where the $f_i$ are homogeneous polynomials?
I wonder how to deduce this from the definition of rational map from On The Arithmetic of Elliptic Curves, Joseph H. Silverman:
Let $V_1$ and $V_2 \subseteq \mathbb{P}^n$ be projective varieties. A rational map from $V_1$ to $V_2$ is a map of the form $$\varphi : V_1 \rightarrow V_2, \qquad \varphi = [f_0,\ldots,f_n]$$
where the functions $f_0,...,f_n ∈ K(V_1)$ have the property that for every point $P ∈ V_1$ at which $f_0,...,f_n$ are all defined, $$\varphi (P) = [f_0(P),...,f_n(P)] \in V_2 \, .$$
Recall that $K(\Bbb P^n)\cong K(\frac{x_1}{x_0},\cdots,\frac{x_n}{x_0})$. Pick $f_0,\cdots,f_n\in K(\Bbb P^n)$ by the quoted definition. Clear denominators and homogenize with respect to $x_0$. This will give you that you may represent your map by homogeneous polynomials in $x_0,\cdots,x_n$.