From the paper, previously you needed a sequence of images from a single night to generate a "tracklet," which lets you estimate a guess of the orbit of the asteroid. The guess can be compared with images from other nights, to confirm the existence and orbit of the asteroids. Without the tracklet, the asteroid could be heading in any direction, meaning that the set of possible orbits is huge, and there would be many many possible matches to other images.
The key insight of THOR (their algorithm) is that even a mediocre guess of the orbit makes the true orbit stand out. From an observed object, they predict a test orbit. Then they take all other images that could be along that orbit, project all objects in each image to the original image according the orbit, and then stack the projected images together. If their guess of the orbit was perfect, there should be a clear dot in the stacked projections, where that single asteroid was seen in all the images. But even an imperfect guess is still useful. Any object that's somewhat close to the guessed orbit will show up as a line in the stacked image. So find the lines, use some math to refine the estimate of that objects orbit, and now you've got the object.
(I'm not in this field, so take all explanations with a grain of salt please).