Consider a superalgebra $A$ and its representation category $\text{Rep }A$. So the objects are assumed to be super-vector spaces and the action should be compatible with the $\mathbb{Z}_2$-grading.
Now, further assume that there is a decomposition of $A$ into ideals $$A=A_0\oplus A_1.$$
What does in this context $$\text{Rep }A=\text{Rep }A_0\oplus\text{Rep }A_1$$ mean?
So I could imagine that objects are of the form $U\oplus V$, where $U\in\text{Rep }A_0, V\in\text{Rep }A_1$. But what are the morphisms?
This statement says that any $V\in \operatorname{Rep}A$ decomposes as $V=V_0\oplus V_1$ with $V_i\in \operatorname{Rep} A_i$ and every morphism $f\in \mathrm{Hom}_A(V,W)$ decomposes as $f=f_1+f_2$ where $f_i\in\mathrm{Hom}_{A_i}(V_i,W_i)$.
The fact that $A$ is a superalgebra is entirely irrelevant to your question.