I would say that understanding traditional differential cohomology is a reasonable prerequisite.
There are multiple good sources:
Ulrich Bunke: Differential cohomology
Diferential Cohomology. Categories, Characteristic Classes, and Connections.
Further suggestions can be found in this answer:
References for differential cohomology and differential characters
For higher topos theory (as it is relevant for DCCT) one could start with
- Daniel Dugger: Sheaves and homotopy theory