What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well aware of several statements made more beautiful in the language of stacks, but, I'm looking for a concrete application.
I would like to point out that stacks are "just" higher analogues of sheaves  a very basic tool to arrange structure. The same is true for topological or differentiable stacks. So I think everybody who expects amazing applications of stacks should be able to name an equally amazing application of a sheaf. (I am not saying that those don't exist!)
That said, let me mention an application. In view of the fact that you didn't get any answers so far (apart from your own), I hope it's not too inappropriate to take one from my own research. It applies abelian gerbes with connection to lifting problems for principal bundles.
I hope the following specifications qualify the theorem below as application: its statement does not involve any stacks or gerbes, just "basic" differential geometry. Its proof, however, is a simple composition of two gerbetheoretical theorems.
Theorem. Let $M$ be a connected smooth manifold, let $P$ be a principal $G$bundle with connection over $M$, let $\hat G$ be a central extension of $G$ by an abelian Lie group $A$, and let $\rho \in \Omega^2(M,\mathfrak{a})$ be a 2form. Then, there exists a principal $A$bundle $\mathcal{L}_P$ over $LM$ with a connection and with a fusion product, and a bijection between
isomorphism classes of lifts of the structure group of $P$ from $G$ to $\hat G$ with compatible connection of scalar curvature $\rho$, and
smooth sections of $\mathcal{L}_P$ that preserve the fusion product and pull back the connection to the transgressed 1form $L\rho \in \Omega^1(LM,\mathfrak{a})$.
Of course some concepts that appear here would need some more explanation  but that's not the point. Let me better point out how gerbes with connection come into the picture. We employ two results from gerbe theory:
Associated to every lifting problem posed by a bundle $P$ is an $A$gerbe over $M$, called the "lifting gerbe" and denoted $\mathcal{G}_P$. This gerbe represents geometrically the obstruction against lifts. Moreover, the actual lifts are in equivalence with trivializations of $\mathcal{G}_P$. The same works if one wants to include connections into the lifting problem. These are results of Murray and Gomi.
The category of $A$gerbes with connection over $M$ is equivalent to a certain category of principal $A$bundles with connection over $LM$ which are additionally equipped with "fusion products". The equivalence is established by a transgression functor, which has been introduced by Brylinski and McLaughlin. It takes trivializations of gerbes to sections of bundles.
Now, define $\mathcal{L}_P$ as the transgression of $\mathcal{G}_P$. Since transgression is an equivalence of categories, it is a bijections on Homsets, and this bijection is exactly the statement of the theorem.
Ok, in order to complete my claim that this is an application, I should probably mention an example where the theorem is useful. That's the case for $spin$ and $spin^c$ structures on manifolds, and I have learned about it from Stephan Stolz and Peter Teichner. In the case of $spin$ structures, $\mathcal{L}_P$ is a $\mathbb{Z}_2$bundle over $LM$ and plays the role of the orientation bundle of $LM$. Since $\mathbb{Z}_2$ is discrete, all the connections disappear and forms are identically zero. So, the theorem says that isomorphism classes of $spin$ structures on $M$ are in bijection to "fusionpreserving orientations" of $LM$. In the $spin^c$ case, a similar statement follows that additionally includes the scalar curvature of the $spin^c$structures.

1$\begingroup$ Sorry for taking so long to reply to this, I fell off of MO during August and forgot to check this. Do you have a reference for this? I'm in the middle of TeXing some gerbe stuff as well... $\endgroup$ – David Carchedi Oct 20 '10 at 10:13

1$\begingroup$ The story is in my paper "A loop space formulation for geometric lifting problems" (arxiv.org/abs/1007.5373). There you also find the references to papers of Murray, Gomi, BrylinskiMcLaughlin and StolzTeicher mentioned above. $\endgroup$ – Konrad Waldorf Oct 20 '10 at 11:10

1$\begingroup$ I guess, saying that differentiable stack (in the sense of, say, nlab) is just a stack is like saying that a manifold is just a sheaf. But I agree: To ask for applications of manifolds to concrete theorems (not mentioning manifolds) might also be a bit artificial. $\endgroup$ – Lennart Meier Feb 5 '14 at 16:16
Whilst asking this, I nearly forgot that one application does come to mind:
http://www.math.fsu.edu/~aluffi/archive/paper325.pdf
In this paper Behrang Noohi shows how to use topological stacks to calculate the fundamental group of the quotient of a topological space by a group(oid) action by using fixedpoint data.
I should update with a mention of some of my own results in http://arxiv.org/abs/1504.02394:
There is a proof of Segal's theorem that the classifying space $B\Gamma^q$ of Haefliger's foliation groupoid is homotopy equivalent to classifying space of the discrete monoid of embeddings of $\mathbb{R}^n$ into itself $B\mathbf{Emb}\left(\mathbb{R}^n\right)$ using differentiable stacks (and higher topos theory). (This is theorem 3.7)
You can also use the same machinery to prove the following theorem (Theorem 4.1):
Let $G$ be a Lie group acting almost freely on a manifold $M$. Then the homotopy type of the Borel construction $M\times_G EG$ is the same as the the classifying space of a certain discrete category, whose objects are smooth tranversals, i.e. maps $f:\mathbb{R}^n \to M,$ with $n=\dim M  \dim G$ which are transverse to the $G$orbits.