There is a standard way of rewriting coequalizers as pushouts: the coequalizer $C$ of $f, f' : X \to Y$ is the pushout
$$\begin{CD} X \amalg X @>(f,f')>> Y \\ @VVV @VVV \\ X @>>> C \end{CD}$$
(where the left vertical map is the fold map, given by the identity on each summand).
Then to see that $C$ is also the cofiber of $f-f'$, look at this diagram:
$$ \begin{CD}
X @>{\begin{pmatrix}1 \\-1\end{pmatrix}}>> X \amalg X @>{\begin{pmatrix}f & f'\end{pmatrix}}>> Y \\ @VVV @V{\begin{pmatrix}1 & 1\end{pmatrix}}VV @VVV \\ 0 @>>> X @>>> C
\end{CD}
$$
Here I've switched to more comfortable matrix notation for morphisms where you implicitly insert the equivalence $\alpha : X \amalg X \cong X \times X$ or its inverse in the morphisms where required. For example, ${\scriptstyle\begin{pmatrix}1 \\ -1\end{pmatrix}}$ means the composite $X \xrightarrow{(\mathrm{id}, -\mathrm{id})} X \times X \xrightarrow{\alpha^{-1}} X \amalg X$.
Notice that the composite across the top is $f - f'$, so if we show that the outer rectangle is a pushout, then $C$ will be the cofiber of $f-f'$.
Since the square on the right is a pushout, to show the rectangle is a pushout it is enough to show the square on the left is a pushout. This is done by using the shear map to turn it into a more obvious pushout square, namely, look at:
$$
\begin{CD}
X @>{\begin{pmatrix}1 \\ 0\end{pmatrix}}>> X \amalg X @>\begin{pmatrix}1 & 0 \\ -1 & 1\end{pmatrix}>> X \amalg X \\
@VVV @V{\begin{pmatrix}0 & 1\end{pmatrix}}VV @V{\begin{pmatrix}1 & 1\end{pmatrix}}VV \\
0 @>>> X @>{1}>> X
\end{CD}
$$
You can check that the outer rectangle is precisely the square we are trying to show is a pushout, so we'll be done if we show each of these two squares is a pushout.
The one on the right is because both horizontal maps are equivalences: the inverse of the top one is the shear map ${\begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix}}$ which is an equivalence in any additive category.
The square on the left is the (pointwise) coproduct of two easy pushout squares:
$$
\begin{CD} X @>>> X \\ @VVV @VVV \\ 0 @>>> 0 \end{CD}
\quad\quad
\begin{CD} 0 @>>> X \\ @VVV @VVV \\ 0 @>>> X \end{CD}
$$
