Tensor product of integral domains over algebraically closed field (Ex. 9.5.O Vakil's FOAG)

You're mixing up $A$ and $A'$ and that's causing your issue. The point is that you can reduce checking whether $A\otimes B$ is an integral domain to checking whether $A'\otimes B'$ is an integral domain where $A'$ and $B'$ are finitely generated over $\overline{k}$, and now you should work with $A'$ and $B'$ by themselves and put $A$ and $B$ out of your mind while you work on this argument. (This is what Johan is doing - he's implicitly assuming $A$ and $B$ finitely generated without saying anything about it in that blog post.)

Once you're reduced to working with finitely generated $A'$ and $B'$, everything is easy enough: for finitely generated algebras over a field, the Jacobson and nil radicals coincide, so if $a_1,c_1$ are in every maximal ideal they are nilpotent and therefore zero since $A'$ is a domain. But they're not, so there's some maximal ideal not containing them. The second statement you're having trouble with is similar: once you know $A'$ is finitely generated over $\overline{k}$, Zariski's lemma shows that $A'/\mathfrak{m}=\overline{k}$ as you discuss in your parenthetical.