The way I understand it, they say that animals react better to danger coming from the left side, because the left visual field (of both eyes) is processed by the right hemisphere, which is dominant for threat processing and spatial attention. So, for the cat sleeping on the left side the danger will probably come from the left side of its visual field, while for cat sleeping on the right side it would come from the right side of the visual field. Therefore, sleeping on the left side is better, because the cat will react faster to something coming towards it.
Look at this picture to see how the image from both eyes is processed:
https://en.wikipedia.org/wiki/Visual_system#/media/File:Huma...
(this is for human visual system, but it's the same for a cat I guess?)
The important thing is not the left/right eye, but the left/right side of both eyes.
That clears it up immensely, appreciate it.
Maybe they never intended for a broad audience, but the paper would be way more accessible if they had included a description like you have here. It was a frustrating read about a well-liked subject, I'm sure I'm not the only one that felt that way.