They seem to be talking about a sort of non-linear absorption of the laser light, so that above some critical intensity that is found only close enough to the focal point of a converging laser beam, something suspended in the resin is triggered to absorb the red and ?? emit blue light, the which causes a chemical reaction in the surrounding resin at just that point, curing it.
Looking at the paper, they talk about red light producing excitons in one material that become triplets of "annihilator excitons" in another bit attached to it, and, with enough intensity, a pair of such triplets that will "fuse" to make a single "hot" exciton, which decays emitting a blue photon. Presumably at lower intensity you get only one triplet at a time which hasn't enough energy to emit the critical blue photon. The suspended particles must be much smaller than the 637 nm red wavelength.
You would still need to build up the object from the inside out, or conventionally bottom up, unless the cured bits pass light exactly the same as the uncured resin. They also seem to suggest that if the resin is especially gelatinous, a cured bit does not move while other bits are being cured, so you don't need the sacrificial support structures.
Presumably, since nothing in the system moves except the light source, you should be able to make this work faster than one that has to add layers of resin.
The bit that catches the red photons seems to be a porphyrin cage around a palladium ion. So you trap some amount of palladium in each article produced. Getting suspended particles to hold together (in a silica shell) but disperse without clumping was its own challenge.