Why is the scheme associated to a product of quasi-projective varieties naturally isomorphic to the product of the associated schemes?

Question

What I mean by this is, suppose $X$ and $Y$ are quasi-projective varieties over some arbitrary field $k$. Then $X\times Y$ is again a quasi-projective variety. I've seen this a few times, but what is the rigorous explanation for why we can make the identification $$\widetilde{X\times Y}\simeq\tilde{X}\times_{\operatorname{Spec}(k)}\tilde{Y}$$ between the associated schemes of $X\times Y$, $X$ and $Y$?