Actually$\DeclareMathOperator\im{im}$Actually I couldn't quite figure out how to do a Ken Brown sort of argument, but here's an argument that works:
Let $L$ denote the usual sheafification functor, a la Grothendieck and Verdier etc. Then I claim it's enough to show $L$ sends local epimorphisms to epimorphisms and local monomorphisms to monomorphisms. Indeed, if this is the case then $L$ takes local isomorphisms to epi-monomorphisms, and such things are isomorphisms in toposes.
Since $L$ preserves finite limits, we need only check that $L$ takes local epis to epis (since $A \to B$ is a local mono iff $A \to A\times_BA$ is a local epi). But that's not so bad: If $A \to B$ is a local epimorphism then the map $im(A \to B) \to B$$\im(A \to B) \to B$ is a local epimorphism and a monomorphism, and hence a monic, local isomorphism. But $L$ preserves images (since they're computed as colimits and L preserves those) and takes monic, local isomorphisms to isomorphisms. Thus $LA \to LB$ has the property that $im(LA \to LB) \to LB$$\im(LA \to LB) \to LB$ is an isomorphism (NB: that image is computed in the category of sheaves), and so the map is epi in the category of sheaves, which is what we wanted.
