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.

For higher topos theory (as it is relevant for DCCT) one could start with

• Daniel Dugger: Sheaves and homotopy theory

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