Specifically, there is a passage in Dummit and Foote that says
Suppose $V$ is a finite-dimensional $FG$-module and $V$ is reducible. Let $U$ be a $G$-invariant subspace. Form a basis of $V$ by taking a basis of $U$ and enlarging it to a basis of $V$. Then for each $g \in G$ the matrix $\varphi(g)$, of $g$ acting on $V$ with respect to this basis is of the form
$\varphi(g) = [[\varphi_1(g); \psi(g)][0; \varphi_2(g)]]$, where $\varphi_1 = \varphi\vert_U$ (with respect to the chosen basis of $U$) and $\varphi_2$ is the representation of $G$ on $V/U$ (and $\psi$ is not necessarily a homomorphism - $\psi(g)$ need not be a square matrix). So reducible representations are those with a corresponding matrix representation whose matrices are in block upper triangular form.
How did they get that matrix?
($\varphi$ is the homomorphism from $G$ to $Aut(V)$, I'm not sure what $\psi$ is)