The basic design is derived from the very animals we got the idea from. All animals that change color do so via a structure called a Chromatophore. These are basically pigment filled sacks and reflective structures that can be manipulated to vary the pigment of the overall tissue.
Currently, we are able to grow a film of chromatophore laden cells on top of an electrode grid. This allows us to stimulate specific regions of the film, much like a LCD screes works. Theoretically, we should be able to stimulate individual 'pixels' and produce a desired color. We aren't quite there yes, but it's only a matter of time until we figure out the proper signaling for each desired pigment. Once that is achieved, the rest of the packaging should be relatively easy to complete.
The only trick left is to develop an effective nutrient and waste transport film. This could potentially be a limiting factor on how large we could make a display, but that's something we'll tackle at another time.