The title essentially explains it, but I'll give some background:

I'm giving a talk to some fellow grad students about the relative Picard functor which requires introducing Grothendieck (pre-)topologies, and I'm curious what the historical reason to introduce them is. Particularly I was trying to think of a natural situation in which one would think of this.

What I came up with was essentially the following: if I have an abelian variety $A$ over a field $k$ and I look at multiplication by $m$ coprime to the characteristic of $k$, I get a natural "exact sequence" of varieties

$ 0 \rightarrow A_m \rightarrow A \overset{m_A}{\rightarrow} A \rightarrow 0$

But the real question here is what the exactness means. If we look at the underlying variety(i.e. closed points) we get an exact sequence of abelian groups, but when we think of the corresponding sequence of presheaves

$ 0 \rightarrow h_{A_m} \rightarrow h_A \rightarrow h_A \rightarrow 0 $

This is no longer exact, instead taking sections we get an exact sequence

$ 0 \rightarrow h_{A_m}(V) \rightarrow h_A(V) \rightarrow h_A(V) $

for any $V/k$. This is starting to look sheafy, and so we hope for some general cohomology theory in which I can write $H^1(V,A_m)$ and get some sort of obstruction to exactness on the right. Grothendieck topologies fill this role, and all is right in the jungle.

Is this at all close to the original reasoning?