TITLE: What are the benefits of viewing a sheaf from the "espace étalé" perspective?
QUESTION [53 upvotes]: I learned the definition of a sheaf from Hartshorne---that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. While fairly abstract at the outset, this seems to be (to me) an intuitive view; in particular, all of the manipulations and constructions with sheaves fit nicely into this schema.
I know that the older view of a sheaf on $X$ was to consider it as a triple
$$
(E, X, \pi)
$$
where $\pi : E \to X$ is a local homeomorphism, and so that the "sheaf of sections" of this map $\pi$ is the sheaf in the functorial sense described above. 
This view makes much less sense to me, but I have to wonder if that is simply due to my having learned it second. However, it also makes me wonder if I am missing something, and so my question is as follows.

What are some (edit:) specific benefits of viewing a sheaf in this sense? What is gained by considering a sheaf as the espace étalé over $X$?

REPLY [39 votes]: To me the obvious answer involves sheafification of a presheaf.  If you look at the construction of the associated sheaf to a presheaf  in, say, Hartshorne it goes through the étalé space construction without specifically telling you, and to me it makes the construction somewhat unmotivated.
Namely, if $P$ is a presheaf on $X$, then taking the stalk $P_x$ at each point of $X$ gives you an $X$-indexed set, or as Tom would say above, a set over $X$.  One can then define a topology on $\biguplus_{x\in X}P_x$ so that the natural projection $\biguplus_{x\in X}P_x\to X$ is a local homeomorphism in the obvious way namely if $U$ is a neighborhood in $X$ and $s\in P(U)$, then $(s,U) = \lbrace germ_x(s)\mid x\in U\rbrace$ is a basic neighborhood.  This topology immediately makes $(s,U)$ homeomorphic to $U$ and makes $s$ a section over $U$ via $x\mapsto germ_x(s)$ for $x\in U$.  The sheaf of sections of $p$ is the associated sheaf of $P$.  I find this construction completely unmotivated without going through étalé spaces.
Added.  Another good reason is it is convenient for defining actions of a topological groupoid on a sheaf.  If $G=(G_0,G_1)$ is a groupoid, a $G$-sheaf is an étalé space $p:X\to G_0$ over $G_0$ together with an action map $G_1\times_{d,p} X\to X$ satisfying obvious axioms.  This is more difficult to phrase in the sheaf as a functor language.  A theorem of Joyal and Tierny says that every Grothendieck topos is equivalent to the topos of sheaves on a localic groupoid.
 Additional additions  From the étalé space point of view it is clear that covering spaces are indeed elements of the topos $Sh(X)$ of sheaves on $X$ and that the fundamental group of $Sh(X)$ (in the sense of Barr and Diaconescu) is the usual fundamental group of $X$ if $X$ is locally simply connected.  
Of course it is not hard to see that covering spaces correspond to locally constant sheaves but I don't think this is the way people think about covering spaces.