Base on the floor first, flat side down. Then run the power cord up through it — before anything goes on top. The cord passes through the middle of the stack, so leaving it until the end means taking the towers off again.
The middle piece drops onto the base and seats with a positive click. The top piece drops onto that the same way. The iPad is in the front of the top piece, and the light switch is on it.
They are keyed — if a piece does not want to go, it is turned around. Rotate it 180° rather than forcing it.
No screws, no latches, nothing to tighten. If all three are seated, the assembly does not wobble.
Two minutes, and you'll know either way.