Let $X \to Y$ be a surjective morphism of sheaves on the site $(\operatorname{Sch}/S)_{et}$ and suppose that $X$ is a scheme. Consider the cartesian diagram $\require{AMScd}$ \begin{CD} F @>{}>> W \\ @VVV @VVV\\ Y @>{\triangle_Y}>> Y \times_S Y \end{CD} where $\triangle_Y$ is the diagonal map and $F:= W \times_{Y \times_S Y, \triangle_Y} Y$ is the fiber product.
Let us assume that $S$ and $W$ are affine schemes.
It is stated in Olsson's Algebraic spaces and stacks in subsection 5.2.9 that: Since $X \to Y$ is a surjective morphism of sheaves, there exists an etale cover $W' \to W$ such that the composition $$W' \to W \to Y \times_S Y$$ factors through $X \times_S X$.
What about the fact that $X \to Y$ is a surjection implies the existence of such an etale cover factoring through $X \times_S X$?
I might be making a mistake here but I think this comes down to the definition of a surjective morphism in the category of sheaves on $(\operatorname{Sch}/S)_{et}$. Namely, $\phi : X \to Y$ is surjective/an epimorphism if for all objects $T \to S$ any element $f \in Y(T)$ can be pulled back to an element of $X(T)$ after possibly taking an etale cover. This means $\exists g: T' \to T$ etale such that the element $f \circ g : T' \to T \to Y$ comes from an element of $X(T')$ under $\phi$. Of course, this is equivalent to $T' \to T \to Y$ factoring through $\phi : X \to Y$. I assume the claim then follows by observing the product map $X \times_S X \to Y \times_S Y$ is also surjective.
See Tag 00WL for definition of surjective morphisms of sheaves on sites.