I still don't understand how he could build an in game computer. What's special about redstone which acts like a real circuit? And how does the computer 'turn' into a computer by putting the redstone in a certain formation?
Redstone can be used to create a Logic Circuit, which is all computers really are down to the bare bones. Think of a light switch in your house. When you press the switch up to turn the light on, a device inside connects the "logic circuit" together which allows electricity to flow to the light bulb. When you press the switch down, the device inside breaks the "logic circuit" and stops the flow of electricity to the light bulb.
In binary terms:
Light on = 1
Light off = 0
Using redstone, and a LOT of time, a logic circuit such as this can be created. Creating a computer within a computer game is the same as the light switch anology but on a MASSIVE scale.
For instance in the video the player places redstone torches in the "input" slots to calculate 1+2. Down the line in his computer are lines and breaks which caused the "output" slots to calculate 1+2=3 or as seen in the video 1,1 which is binary for 3.