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The treatment in EGA is indeed intimidating, but in fact over a field the formula is not hard to prove. You only need $X$ and $Y$ to be separated schemes over $k$, and $\mathcal{F}$ and $\mathcal{G}$ to be quasi-coherent. Then cover $X$ and $Y$ by affine open subsets $\{U_i\}$, and $\{V_j\}$, and write down the Čech complex for $\mathcal{F}$ and $\mathcal{G}$ with respect to these two coverings, and the Čech complex of $\mathcal{F} \boxtimes \mathcal{G}$ with respect to the covering $U_i \times V_j$. It is not hard to see that the last is the tensor product of the first two; then the thesis follows from Eilenberg-Zilber (or however you want to call the fact that the cohomology of the tensor product of two complexes over a field is the tensor product of the cohomlogies).