The claim is:
__This battery holds 10x more charge per kilo of material than previous attempts at STRUCTURAL (massless) battery materials__.
To be specific, this material holds only _ONE FIFTH_ the charge of what your smartphone's battery can hold per kilo of battery. No laws of thermodynamics is being broken and this is not at all about battery chemistry. It's all about the 'physics' of construction materials: About how this stuff is made in the factory and layered. The basic battery chemistry going on is not much different from what's been available for years - the interesting part is how this material encases it.
You can't make a car by building the chassis out of smartphone batteries. But the promise of this paper is that you CAN build the car chassis out of this battery, and even if this battery is only 20% as effective, a car chassis is rather large, and you needed it anyway, so every drop of power you can store in the chassis itself was effectively 'free' - hence the somewhat hyperbolous 'massless' terminology.