If $W$ is a $G$-invariant proper subspace of the finite dimensional vector space $V$, why do we have
$$G \le \Bigg\{ \begin{pmatrix} \begin{array}{c|c} *&* \\ \hline 0&* \end{array} \end{pmatrix} \Bigg\},$$
and how do we know every $g \in G$ can be written as:
$$\begin{pmatrix} \begin{array}{c|c} \varphi(g) & *\\ \hline 0 & \psi(g) \end{array} \end{pmatrix},$$
where $\varphi:G \to GL(W)$ and $\psi:G \to GL(V/W)$ are the natural homomorphism induced by the $G$-invariance of $W$?
There's nothing fancy going on here - the unipotence assumption is a red herring. Just select an ordered basis for $W$, and complete it to a basis for $V$, and use that basis to obtain a $G$ as a subgroup of the group $GL_n$ of invertible matrices. The fact that $W$ is an invariant subspace gives you that the lower-left block in your picture is automatically $0$.
Then $\varphi$ is a matrix form of the action of $G$ on $W$, because that's what you get when you restrict the action of $G$ to $W$.
The lower-right block is a matrix form of the action of $G$ on $V/W$; when you quotient out by $W$, you restrict to the second column of your picture, and ignore the entries above the lower-right block. The resulting matrix form is exactly $\psi$.