[proofplan]
We verify the biproduct identities in $\mathcal D$ for the four morphisms obtained by applying $F$ to the biproduct structure maps in $\mathcal C$. Functoriality preserves identities and composites, while additivity preserves zero morphisms and addition in each hom group. Applying these two properties to the biproduct equations for $X=A\oplus B$ gives exactly the biproduct equations for $F(X)$.
[/proofplan]
[step:Apply functoriality to the diagonal projection-injection identities]
Let $X := A \oplus B$. Since $X$ is a biproduct of $A$ and $B$ in $\mathcal C$, its structure maps satisfy
\begin{align*}
p_A \circ i_A &= \operatorname{id}_A, &
p_B \circ i_B &= \operatorname{id}_B.
\end{align*}
Applying the functor $F:\mathcal C \to \mathcal D$ and using preservation of composition and identity morphisms, we obtain
\begin{align*}
F(p_A)\circ F(i_A)
&= F(p_A\circ i_A)
= F(\operatorname{id}_A)
= \operatorname{id}_{F(A)},\\
F(p_B)\circ F(i_B)
&= F(p_B\circ i_B)
= F(\operatorname{id}_B)
= \operatorname{id}_{F(B)}.
\end{align*}
[/step]
[step:Use additivity to send the off-diagonal composites to zero]
The off-diagonal biproduct identities in $\mathcal C$ are
\begin{align*}
p_A\circ i_B &= 0_{B,A}, &
p_B\circ i_A &= 0_{A,B},
\end{align*}
where $0_{B,A}:B\to A$ and $0_{A,B}:A\to B$ denote the zero morphisms in $\mathcal C$. Since $F$ is additive, for every pair of objects $Y,Z\in\operatorname{Ob}(\mathcal C)$ the induced map
\begin{align*}
F_{Y,Z}:\operatorname{Hom}_{\mathcal C}(Y,Z)&\to \operatorname{Hom}_{\mathcal D}(F(Y),F(Z))\\
f&\mapsto F(f)
\end{align*}
is a homomorphism of abelian groups. Therefore it sends zero morphisms to zero morphisms. Hence
\begin{align*}
F(p_A)\circ F(i_B)
&= F(p_A\circ i_B)
= F(0_{B,A})
= 0_{F(B),F(A)},\\
F(p_B)\circ F(i_A)
&= F(p_B\circ i_A)
= F(0_{A,B})
= 0_{F(A),F(B)}.
\end{align*}
[/step]
[step:Preserve the identity decomposition by additivity on the endomorphism group]
The remaining biproduct identity in $\mathcal C$ is the identity decomposition in the abelian group $\operatorname{End}_{\mathcal C}(X)$:
\begin{align*}
i_A\circ p_A+i_B\circ p_B=\operatorname{id}_X.
\end{align*}
Both $i_A\circ p_A$ and $i_B\circ p_B$ are endomorphisms of $X$, so additivity of
\begin{align*}
F_{X,X}:\operatorname{End}_{\mathcal C}(X)&\to \operatorname{End}_{\mathcal D}(F(X))\\
f&\mapsto F(f)
\end{align*}
gives
\begin{align*}
F(i_A\circ p_A+i_B\circ p_B)
=
F(i_A\circ p_A)+F(i_B\circ p_B).
\end{align*}
Using functoriality for the two composites and the identity decomposition in $\mathcal C$, we get
\begin{align*}
F(i_A)\circ F(p_A)+F(i_B)\circ F(p_B)
&=F(i_A\circ p_A)+F(i_B\circ p_B)\\
&=F(i_A\circ p_A+i_B\circ p_B)\\
&=F(\operatorname{id}_X)\\
&=\operatorname{id}_{F(X)}.
\end{align*}
[guided]
The only identity that uses more than ordinary functoriality is the decomposition of the identity. The morphisms $i_A\circ p_A:X\to X$ and $i_B\circ p_B:X\to X$ lie in the same hom group, namely $\operatorname{End}_{\mathcal C}(X)$, so their sum is defined there. Because $F$ is additive, the map on this hom group,
\begin{align*}
F_{X,X}:\operatorname{End}_{\mathcal C}(X)&\to \operatorname{End}_{\mathcal D}(F(X))\\
f&\mapsto F(f),
\end{align*}
is a homomorphism of abelian groups. Thus it preserves addition:
\begin{align*}
F(i_A\circ p_A+i_B\circ p_B)
=
F(i_A\circ p_A)+F(i_B\circ p_B).
\end{align*}
Functoriality then rewrites each summand as a composite in $\mathcal D$:
\begin{align*}
F(i_A\circ p_A)&=F(i_A)\circ F(p_A),\\
F(i_B\circ p_B)&=F(i_B)\circ F(p_B).
\end{align*}
Since the original biproduct structure in $\mathcal C$ satisfies
\begin{align*}
i_A\circ p_A+i_B\circ p_B=\operatorname{id}_X,
\end{align*}
we conclude
\begin{align*}
F(i_A)\circ F(p_A)+F(i_B)\circ F(p_B)
&=F(i_A\circ p_A)+F(i_B\circ p_B)\\
&=F(i_A\circ p_A+i_B\circ p_B)\\
&=F(\operatorname{id}_X)\\
&=\operatorname{id}_{F(X)}.
\end{align*}
This is the identity decomposition required for $F(X)$ to be a biproduct object.
[/guided]
[/step]
[step:Identify the resulting structure as a biproduct in $\mathcal D$]
We have proved the four projection-injection identities
\begin{align*}
F(p_A)\circ F(i_A)&=\operatorname{id}_{F(A)},&
F(p_B)\circ F(i_B)&=\operatorname{id}_{F(B)},\\
F(p_A)\circ F(i_B)&=0_{F(B),F(A)},&
F(p_B)\circ F(i_A)&=0_{F(A),F(B)},
\end{align*}
and the identity decomposition
\begin{align*}
F(i_A)\circ F(p_A)+F(i_B)\circ F(p_B)=\operatorname{id}_{F(X)}.
\end{align*}
These are precisely the biproduct equations in the additive category $\mathcal D$ for the object $F(X)$ with injections $F(i_A),F(i_B)$ and projections $F(p_A),F(p_B)$. Therefore $F(X)=F(A\oplus B)$ is a biproduct of $F(A)$ and $F(B)$ in $\mathcal D$ with the stated structure maps.
[/step]
