I think the idea here is to bake it into construction that would happen anyway. If you just need energy storage, keep using batteries. But if you're pouring a foundation already, why not also turn that foundation into a battery?

concrete seems to be used here for its structural properties, just like we do it today. Their solution doesn't seem to require it:

If more powerful capacitors are required, they can be made with a larger concentration of carbon black, at the expense of some structural strength. This could be useful for applications where the concrete is not playing a structural role or where the full strength potential of concrete is not required. For applications such as a foundation, or structural elements of the base of a wind turbine, the “sweet spot” is around 10 percent carbon black in the mix, the team says.