Toposes were invented to define étale cohomology (and with that, prove the Weil conjecture). All this is written down in SGA4 and SGA$4\frac{1}{2}$.
However, it can be a bit overwhelming, to try to read these sources.
Question: Are there other resources with which one can learn the theory of (étale) cohomology of Grothendieck toposes?
Often recommended books in this context are Milne's Etale cohomology, Artin and Mazur's Etale homotopy, and Artin's lecture notes Grothendieck topologies, but all these sources don't directly discuss toposes as far as I can see. I'd like to see an introduction directly discussing toposes.
You need to decide whether you want to study etale cohomology or topos theory --- except for SGA 4, they are rather distinct topics. For example, SGA 4.5, which is Deligne's exposition of the prerequisites on etale cohomology needed for his proof of the last of the Weil conjectures doesn't mention them as far as I know. Etale cohomology was developed for its applications to algebraic geometry, and doesn't need any topos theory. Topos theory is part of logic and the foundations of mathematics.