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.