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 etale space construction without specifically telling you and to me it makes the construction somewhat unmotivated.
Namely, taking 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 $P$. If find this construction completely unmotivated without going through etale 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 etale 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 etale 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.
