PShimi

Just like the movie Red Planet (2000) - they had rollable screens then. It was pretty cool.

Ah! Just read US Patent No 10,602,623, interesting. 
What exactly are Apple patenting here? It seems - a display that can fit in a hinge, how it would bend, and how the layers of the display would be kept in place and not buckle.

Essentially they are using a flexible display semi laminated to the inner surface of a transparent material, the gap filled with material (maybe even a gel) with the correct refractive index that it would make the display look closer to the surface.
Imagine a book - this would be used on the spine. With two standard display panels on the front and back cover.
The spine display integrates with the front and back covers such that you can's see the join between the displays.
Now, if you have a phone like this, and were to open the phone like a book (only you look at the cover of the book instead of inside the book) you would now have a tablet.

The grooves cut into the material are so that it can bend. That's standard, nothing new.

I didn't see the rollable screens from Red Planet in this patent though.