Ok, so, I will try to answer this best I can. first, I'll tell you a skewed-perspective of what an infinity topos is (or ought-to-be). As for how you can "do differential geometry"- this is a bold statement. But, there's certain aspects of differential geometry which extend naturally / have a nice interpretation in (certain) infinity topoi (with extra structure).

What is an infinity topos? It certainly can wear many different cloaks, and I won't attempt to give a global overview; there's reason that even solid books on $1$-topoi are quite long, since topoi can be thought of as generalized spaces, or theories in logic, or universes that behave like Set, etc.. I will concentrate on just one particular aspect of infinity topos theory. You may have heard the slogan "a topos is a category that behaves like the category of sets". In this vain, the analogous slogan is "an infinity topos is an infinity category that behaves like the infinity category of spaces (thought of as homotopy types, i.e. infinity groupoids)." Basically, an infinity topos provides one with a place in which to do homotopy theory. Objects in an infinity topos have homotopy groups, you can talk about Eilenberg-Mac Lane objects etc. To be more concrete, just as a 1-topos arises by taking sheaves of sets on a 1-category, an infinity topos arises as taking "sheaves" of spaces (infinity groupoids) over an infinity category. (Although, the concept of sheaf is different *AND* to be more correct, every infinity topos arises as a so-called cotopological localization of an infinity category of infinity sheaves on an infinity category- this is due to the failure of Whitehead's theorem internal to the sheaf topos) Basically, one should think of an infinity topos of sheaves (or some hypercompletion thereof etc.) on a category $C$ to be some sort of hybredization of "generalized objects of $C$" and infinity groupoids. What do I mean? Well, sometimes people view (at least certain) sheaves on the category of manifolds as generalized manifolds. In fact, you can faithfully represent all infinite dimensional manifolds this way. What is an example of something between a manifold and a groupoid? An orbifold. Orbifolds (and more general differentiable stacks) naturally lives in the 2-topos of sheaves (stacks) of groupoids on manifolds. An orbifold / differentialble stack is like a manifold whose points can posses finite / Lie intrinsic symmetry groups. What if these symmetry groups themselves didn't form a manifold, but another differentiable stack? Then you would be looking at a higher differentiable stack (this one would live in the 3-topos). In general, everything lives in the infinity topos of sheaves of infinity groupoids on manifolds.

Ok, what does this have to do with principal bundles? If you're given a Lie group $G,$ one can consider this as a group object in manifolds, hence a groupoid object in manifolds (with one object), and hence a (representable) sheaf of groupoids on manifolds. It's not a stack though, but the stack it represents, is the functor $$Mfd^{op} \to Gpd$$ sending a manifold $M$ to the groupoid of principal $G$-bundles over $M$. This stack, is often denoted by $BG$ and is quite formally related to the topological classifying space (and in fact, as a differentiable stack, has the same homotopy type as this). By the Yoneda lemma, maps (not homotopy classes, ALL maps) from $M$ to $BG$ are the same as principal $G$-bundles on $M,$ and $BG$ caries a universal principal $G$-bundle over itself, just as in the topological picture. Now, suppose you cared about line bundles, then you could let $G=U(1)$ and then $BU(1).$ Suppose instead, you cared about bundle-gerbes, then you take $B^2U(1),$ which corresponds to viewing $U(1)$ as a $2$-groupoid with one object, and for bundle $2$-gerbes $B^3U(1),$ etc.

Ok, but how is this differential geometry? Right, so, it isn't. Not yet. I don't think it's accurate to say you can "do differential geometry" in an infinity topos. What is true though, is that there are many interesting infinity topoi *with extra structure* which in addition to having a notion of principal bundle etc., have a good notion of principal bundle *with connection* and allow you make good sense of "differential cohomology". (You can also make sense of things like "higher" Lie theory etc.). What Urs noticed is, you can define all these things inside any infinity topos admitting a "cohesive structure". I don't expect the definition to be enlightening. The point is, inside any "cohesive infinity topos" one can make sense of principal bundles with connection, and more generally, basically everything you need to make sense of so-called "higher gauge theory" (Urs' motivation comes from physics). E.g., the language lets you make good sense of what is a smooth $String(n)$-bundle with connection- or what a smooth $FiveBrane(n)$-bundle is, with connection. This is what Urs means when he says you can "do differential geometry."

Anyhow, I hope Urs himself chimes in as well.