I am having some difficulties proving the following statement:
If $V$ and $W$ are vector spaces over a field $F$ such that $V=V_1\oplus V_2$ and $W=W_1\oplus W_2$ for some subspaces $V_i\subseteq V$ and $W_i\subseteq W$ for $i=1,2$ and $A=\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}$ then $A_{ij}=\Gamma_{W_i}A\Gamma_{V_j}:V_j\to W_i$ for $i,j=1,2$.
Here $\Gamma_{V_j}$ is the projection of $v\in V$ onto $V_j$ along $V_i$ and similarly for $\Gamma_{W_i}$.
I have examined the behavior on an element of $V$ using the relation $Av=w$ but have been unable to establish the necessary equality. Any advice would be greatly appreciated.
Your statement as written is incorrect. We have $$ A_{ij} = \Gamma_{W_i} A \iota_{V_j}, $$ where $\iota_{V_j}$ is the inclusion map from $V_j$ to $V$.
I assume that $\oplus$ denotes an external direct sum, which is to say that we can write each $v \in V$ as a block vector $(v_1,v_2)$ with $v_i \in V_i$.
For any $v \in V_2$, we have $$ \Gamma_{W_1} A \Gamma_{V_2} v = \pmatrix{A_{11} & A_{12}\\ A_{21} & A_{22}} \pmatrix{0\\v} = \Gamma_{W_1}\pmatrix{A_{12}v\\ A_{22} v} = A_{12}v. $$ Since $\Gamma_{W_1} A \Gamma_{V_2} v = A_{12}v$ for all $v \in V_2$, we conclude that $\Gamma_{W_1} A \Gamma_{V_2} = A_{12}$.
The general proof for $W_i,V_j$ is similar.