Aerogels formerly held the record for least dense solid. They're not gels, but they're called aerogels because they're made from gels. As the name "aero" implies, they're mostly air, giving them incredible strength-to-weight ratios and insulating properties. Silica aerogels are translucent and quite amazing to hold; it's like holding a cloud.
Aerogels are made from colloids (a.k.a. gels), which are long chains of polymers formed in water. Free-floating around in the water, they form very long, intricate, 3-D maze-like structures. If you've ever made JELL-O, you've made (and eaten) a colloid. It’s the intricate structure of the colloid which keeps the suspended water from spilling out.
Silica aerogels are formed by removing the water from a silica gel, leaving only the maze like structure behind. This structure is very delicate, and if you attempt to evaporate the water out near room temperature/pressure, the capillary action of water will collapse the structure like a dried out jellyfish. However, if you heat and pressurize the system past the critical point, the water becomes a supercritical liquid and it can be removed without pulling the rest of the material inwards.
It appears that what they’ve done here is similar, except they use a multi-step process to deposit carbon on the colloid, then remove the colloid completely, leaving nothing but a carbon maze. I presume they can't make colloids out of carbon directly, hence the multistep process.