I haven't read all of DCCT so take this with a grain of salt, but after having spent a lot of time with it, this is how I would recommend getting started on the abstract stuff.
First one must learn classical Grothendieck Topos Theory. Chapter 1.2 of DCCT gives a pretty good motivation and some nice examples of sheaves on $\mathsf{Cart}$, but I would recommend David Carchedi's course on Topos theory, which is the quickest course that I could find that covered most of the relevant material.
After learning Grothendieck topos theory one must then learn how to combine homotopy theory with topos theory, this was originally achieved using simplicial sheaves or presheaves, and then was abstracted to the definition of a model topos. For this there is
- Jardine's book Local Homotopy Theory (only the first few chapters are really necessary),
- Dan Dugger's papers: Hypercovers and Simplicial Presheaves, Weak Equivalences of Simplicial Presheaves, Universal Homotopy Theories, and his unfinished expository paper that Dmitri Pavlov linked to.
- Toen & Vessozi's paper,
- Rezk's notes on model topoi, and this master's thesis on model topoi. These presentations are often used to provide examples of objects in higher topos theory.
- I'd also recommend these wonderful notes from Dmitri Pavlov's class. I think I would actually start here.
Now one must learn Infinity Topos Theory. This is harder to recommend resources for as there are much fewer. There are many places to find recommendations for resources on learning infinity category theory, but honestly you don't need to delve into too much of the details to understand much for DCCT, you can really take much of infinity category theory as a black box that just works like usual category theory but with equivalences instead of isomorphisms and homotopy type mapping spaces instead of hom sets. I'd recommend Rezk's notes on quasicategories, get comfortable with the basics, and then watch Rezk's lectures on Youtube, as well as Joyal's.
I'm not saying you need to look at all of these resources, but what you need to know is contained in them. I would start off by first looking at some of Schreiber's previous papers that he mixed into DCCT, like Cech Cocycles for Differential Characterstic Classes and the Principal Infinity Bundles papers 1 and 2.
Addendum: Morally, the use of higher topos theory in DCCT is as a generalization of the nonabelian cohomology of Grothendieck and Giraud. In DCCT, these generalized cohomology classes are given by higher principal bundles. The use of higher category theory makes the formulation of this theory very elegant, but ultimately it is grounded in the theory of stacks and gerbes. Knowing this theory is not necessary for understanding DCCT or HTT, but it is a great way to build motivation and see what a lot of this theory is actually being used for. Here are some references in the differential geometry setting:
- I'd start with Noohi's really short notes,
- Ginot's notes,
- Behrand and Xu - Differentiable Stacks and Gerbes,
- and probably the most relevant to DCCT's formalism is Carchedi's thesis.