The box modifications will be fairly simple. First, unscrew the lid from the hinges if possible.
If you're using hand tools, measure and mark windows slightly smaller than the LCD panels onto the box lid. Otherwise, if using CNC or laser cutter, load the CAD file into your software and let the robot do you work.
Either way, the template below may be helpful.
Here's the lid after cutting out the windows on a laser cutter.
To build the shadowbox chambers that sit below the windows, you can use similar techniques and tools to cut out the slots from any material you like. Darker materials will work best, such as black cardstock, painted cardboard, stained wood, or dark colored plastic. I used black acrylic.
Again, here's a handy template! You may need to adjust depending on your box's dimensions.
Set the long slots in first, then add the shorter ones to form the boxes.