Here is an elegant application of sheaves seen as étalé spaces.

Consider a complex manifold $M$. It automatically comes with a holomorphic local isomorphism $\pi: \mathcal O_M \to M $ described as follows. As a set $\mathcal O_M$ is the set of all germs of holomorphic functions at all points of $M$.The map $\pi$ sends a germ to the point at which it is considered. Then we endow $\mathcal O_M$  with the following topology. For an open connected set $U\subset M$ and a holomorphic function $f$ on $U$, denote by $[U,f]\subset\mathcal O_M$ the set of all germs $f_a$ with $a\in U$.
These $[U,f]$ are decreed to be an open basis for the topology of $\mathcal O_M$.Then there exists a unique complex structure on $\mathcal O_M$ such that $\pi: \mathcal O_M \to M $ becomes a HOLOMORPHIC local isomorphism. On $\mathcal O_M$ there lives a universal tautological holomorphic function
 $F:\mathcal O_M \to \mathbb C: f_a \to f_a (a)$. (Note that $\mathcal O_M$ is huge, disconnected but Hausdorff). 

And now for the punchline : given a holomorphic function $f$ on $U\subset M$, take the connected component $Riem(U)$ of $[U,f]$ in $\mathcal O_M$. Together with the restriction $F|Riem(U)$,  this is the maximal holomorphic extension of $f$: a sophisticated concept admirably handled by sheaves as étalé spaces  (The manifold $Riem(U)$ is called the domain of existence of $f$.) 

Even in dimension one and for $M=\mathbb C$ this is quite powerful: you get the Riemann surface $(Riem(U), F|Riem(U))$ of any holomorphic function $f$ on an arbitrary domain $U\subset \mathbb C$
without the cutting, pasting, continuation along paths,... of which classical books on complex analysis are so fond.

A reference for this might be Fritzsche-Grauert's book "From Holomorphic Functions to Comples Manifolds", Chapter II, $$8,9 (Springer, GTM 213). The book by Narasimhan and Nievergelt that Charles so pertinently and quickly evoked  seems to handle the dimension one case (which actually suffices to convey the sheaf idea).

Finally, it is noteworthy that the EGA-style definition that Hartshorne gives for the structure sheaf $\mathcal O$of the affine scheme $Spec(A)$ (page 70 of THE BOOK) is exactly analogous to the description above: the étalé space is the disjoint union of the all the local rings $A_P$ for $P\in A$ and $\mathcal O(U)$ is the set of continuous maps of $U$ into the étalé space;
Only, Hartshorne doesn't say what the topology is on the étalé space and the continuity condition is replaced by an ad hoc description in terms of elements of the rings of fractions $A_f$.