It's not clear what causes it to solidify, though they say in the video that it does so when it hits the outside air.
You can use air or nitrogen in your tires.
From the FAQ: (no explanation)
When penetration occurs, the tire pressure forces PermaSeal into the wound area sealing instantaneously around the penetrating object. If the object self extracts then, as the tire rotates the tread area is deformed when in contact with the road surface, the puncture wound is forced open and the product is forced into the puncture wound by the air pressure within the tire.