Let $f\colon X \rightarrow Y$ be a proper morphism of schemes, $Y$ noetherian (I don't know which of those assumptions is actually needed for the claim).
According to my lecture notes, for $\mathcal{O}_X$-modules $\mathcal{F}, \mathcal{G}$ there exists a canonical map of higher direct images
$R^qf_{\ast}\mathcal{F} \otimes R^pf_{\ast} \mathcal{G} \rightarrow R^{p+q} f_{\ast}(\mathcal{F} \otimes \mathcal{G})$
How is this map defined/does it arise?
Do you want a fancy, or a non-fancy answer?
If you want a non-fancy answer it suffices to construct a map of presheaves
$$A^pf_\ast\mathcal{F}\otimes A^qf_\ast\mathcal{G}\to A^{p+q}f_\ast(\mathcal{F}\otimes\mathcal{G})$$
where $A^i f_\ast\mathcal{M}$ (the notation of which I just made up, it's not standard), for a quasicoherent module $\mathcal{M}$, is the one sending $U\to H^i(f^{-1}(U),\mathcal{M})$. So now, it's clear want to do. You have a bilinear map
$$H^p(f^{-1}(U),\mathcal{F})\times H^q(f^{-1}(U),\mathcal{G})\to H^{p+q}(f^{-1}(U),\mathcal{F}\otimes\mathcal{G})$$ coming from the functoriality of cohomology. Alternatively, if you assume that your schemes separated and quasicompact, it's clear how to construct the map on Cech cycles, with affine open covers.
This map then gives a map of presheaves
$$A^pf_\ast\mathcal{F}\otimes A^qf_\ast\mathcal{G}\to A^{p+q}f_\ast(\mathcal{F}\otimes\mathcal{G})\to R^{p+q}f_\ast(\mathcal{F}\otimes\mathcal{G})$$
where the last map is the canonical map associated to a sheafification. Then, by the universal property of sheafification, we obtain a map
$$R^p f_\ast\mathcal{F}\otimes R^qf_\ast\mathcal{G}\to R^{p+q}f_\ast(\mathcal{F}\otimes\mathcal{G})$$
as desired.
There may be a fancier way not needing to pass through the presheaf.