I've read that toposes are extremely important in modern mathematics, but I find the definitions and examples given on the nLab page a little too abstract to understand.
Can you provide some examples of toposes which are of use to analysts? e.g., toposes where the objects consist of topological spaces, measure spaces, etc.
First realize that the right definition of orbifold is build on topos (the definition without topoi are mostly cheating).
Here is an example where similar things appear as limit of Riemannian manifold. [For more info, check "On the long-time behavior of type-III Ricci flow solutions." by Lott or the appendix in paper "Diffeomorphism Finiteness, Positive Pinching, and Second Homotopy" with Tuschmann.]
Consider the circles $\mathbb S^1_\ell$ of length $\ell>0$. As $\ell\to 0$, in Gromov--Hausdorff topology they converge to point. On the other hand, the space of harmonic 1-forms stay the same on $\mathbb S^1_\ell$, so for some problems Gromov--Hausdorff topology is not right choice.
You may think of $\mathbb S^1_\ell$ as $\mathbb R/(\ell{\cdot}\mathbb Z)$. Then it is natural to think that $\mathbb R/\mathbb R$ is the limit of $\mathbb S^1_\ell$; More precisely $\mathbb R/\mathbb R$ has one chart $\mathbb R$ and all the shifts $x\mapsto x+a$ as the transition maps. So the limit is smooth, but not a manifold. You may think of it as a pseudogroup of transformations, but you need topoi to give an invariant definition (the same story as with orbifolds).
There are many examples.
You might like Rousseau's reduction of many complex variables to a single one.
In general, topos theory allows one to avoid the axiom of choice and classical logic. For instance, see our concrete treatment of parts of the theory of Banach algebras.
These ideas of course have their roots in the work of Banaschewski and Mulvey of finding a proper treatment of sheaves of C*-algebras/ C*-bundles by considering a single C*-algebra in a sheaf topos. This work is also used in the Bohr toposes mentioned above.
Terence Tao's cheap nonstandard analysis can be interpreted as taking place in a topos related to the topos $\text{Set}^{\mathbb{N}}$ of sets indexed by the natural numbers; see this math.SE question for details.
You should Google for "synthetic differential geometry".
Besides the important examples of topoi already mentioned (like SDG), I would argue that the most important topos for any analyst is just the topos of sheaves on some topological space. I assume that the statement "sheaves are important for analysts" is well-known and doesn't require further argumentation. The study of topoi naturally focuses our attention not on model-dependent results, bizarre axioms and exceptional cases, but on general phenomena and parameter-dependent versions of statements, since the objects of a sheaf topos are naturally dependent on a point in the base space. For these reasons topos theory advocates constructive approach to mathematics, not because of some philosophical reasons, but because logical and set-theoretic intricacies simply don't make sense once you start working over a general base. For example, the axiom of choice fails unless our topological space is discrete. Another example: if you want to do some parametrized topology, you really should consider moving from topological spaces to locales, since general topological spaces over a base are very, very badly behaved.
From this point of view topos theory is some abstract machinery that allows you to transform a good enough (i.e. constructive) proof of any theorem into its parametrized version.
Things get even more fun once you become interested in some homotopical phenomena and deformation theory. It can become very tricky to prove something without considering $\infty$-topoi, either explicitly or shyly hiding them.
I don't know if it counts for "analysis", but toposes (=topoi) appear to be related to quantum mechanics. There's even a book on the subject. For a specific example, a keyword might be Bohr topos.
The point of talking about topoi is rarely to prove something new or interesting about a particular category of spaces. Quite the opposite, the point is typically to prove something interesting about topoi which applies to all categories of spaces that also happen to be topoi. The point is that the abstraction is powerful enough to prove quite a bit for many interesting categories.
Categories of Banach ($\textbf{Ban}$ of all and $\textbf{Ban}_{fd}$ of finite dimensional) and Hilbert spaces ($\textbf{Hilb}_{\oplus}$, $(\textbf{Hilb}_{\oplus})_{fd}$, $\textbf{Hilb}_{\otimes}$, and $(\textbf{Hilb}_{\otimes})_{fd}$) (and related spaces of interest to analysis) can be seen as symmetric monoidal categories, with additional structure. Like topoi, monoidal categories are important and powerful structures with many important theorems proven in them.
These categories are Galois adjoint to topoi in ways that allow natural internal logics to be built and find consonance with the adjunction. This is one way in which topoi are seen in relation to these symmetric closed monoidal categories. You see this association, for instance, in operationalist reductions of quantum theories (which are formulated over a category of Hilbert spaces).
But although these adjoints have some structure-preserving properties, I have found that many treatments stick with the natural categorial structure and prove things in the natural internal logics. Braided monoidal categories have a huge amount of structure and really, I think many will agree that two of the most important types of categories to study when learning categorial methods are topoi and monoidal categories, as that already gets a good part of modern mathematics.
However, if you are interested in topos theoretical direction exclusively, another direction important in analysis besides that above and those of other answers is Pestov's Conjecture, that there exists a Grothendieck topos whose internal Banach spaces are operator spaces. This direction of research is intended to provide a natural category that shows which theorems on Banach spaces extend to operator spaces in general.