First, they claim that the presence of multiple limbs, heads, etc. in cave paintings gave the sense of animated movement when illuminated by the flickering of torchlight. In Reinventing Comics, Scott McCloud argued that this multiplicity in cave paintings implied motion as well, though he didn't jump all the way to claiming it was animation (as in the video below).
I definitely believe that this multiplicity would have implied motion, but the real question is whether it would be fully "animated" by the flicker of torchlight, as they seem to argue. I could understand how torchlight would give the sense of a strobe light, allowing a person to shift attention to different parts of the image, thereby giving the sense of motion.
Nevertheless, I would like to see a video that at least simulates this to really believe it fully. The movie above that makes the case for this nicely shows how the pieces of the pictures could create this effect. However, the movie deletes portions of the image at every step in the animation. A horse with multiple heads on the wall wouldn't do this. It would simply have all the heads all the time, and a viewer would have to try to shift their focus to the other parts throughout the flickering of light.
That's not to say that the animation effect can't happen under these conditions. I'd just like to see a "torchlight" demonstration before I believe it fully.
Their other example I feel is much more compelling though. They claim to have evidence of early "thaumatropes" made of bone, which are small discs that show figures on either side. When spun with a string, they create the illusion of motion (like this animated gif). Toys like these were popular in the Victorian era, but they claim these prehistoric individuals came up with them thousands of years prior.
These seem a lot more probable as early animation, especially given their evidence that the bone discs had figures on both sides and holes in the middle where string could have been placed. The fact that these more convincing devices supposedly accompanied the cave walls perhaps gives more validity to the animation on the wall paintings as well.
At the very least, I take these examples to be far more credible than other examples of ancient animation that have been reported over the years.