It seems, from the example, that they do not write in discs, but in arcs. That is, they don't really spiral in to reach the center.
So I pondered: why would this work, and for whom?
Eyes are a possible reason that it might make for a better form of writing: a species with differential focus/zoom (like an eagle, with the center of the eye very zoomed in, but the outer eye more wide-angle), and good peripheral vision, might find it easier to focus with its wide-angle view, especially if moving the eyes is strenuous. So, writing going in a circle around the periphery of its vision would be ideal.
But what mechanism would create such a form of writing?
What about a very stretchy latex-like tube, coarse on the inside but teflon-smooth on the outside? Thread it through the back of a large ring, and stretch the opening of the tube back over the ring, so it's wrapped around the ring.
You can then read from, or write on, the part of the tube stretched out on the ring.
Squeezing the tube that wrapped over behind the ring will pull more tube through, until eventually the whole tube is inside out - you then need to undo the operation to "rewind" the book. Or maybe just the page: whether to have a whole book on a single coil, or a book made of a sheaf of coils is up to you (or to the publisher). Sort of the difference between long old-world scrolls, and newer books with short pages.
With one tube per page, this would end up looking not unlike the Incan Khipu knot-writing, but with far greater information-density, and obviously no knots (though a knot would work as a "spoiler" tag, you'd have to undo it before reading; and a knot sealed with a chop would work for privacy protection):
Historically, this would have been a good way to store very dense writings in a very small space, because the text would shrink as the surface did - a single coiled hose could hold a whole encyclopaedia! Also, the writings would be protected on the inside of the hose.
In modern times, of course, devices would be just a ring, with a display on, and no hose... much more convenient, but some people would feel that it wasn't the same as curling up with a good coil.