Having now made one, I can give the following evaluation:
It IS a trick. It goes back to what I mentioned earlier: TOLERANCES ADD UP. I used to work with real engineers and that's one of the first things I learned. No matter how it looks, those tolerances are still in there.
The model I made was cut out of a larger square of cardboard, mounted on a thick pasteboard backing. There is 1 1/2" cardboard border. It's all one piece so it won't change size.
In the first (original) configuration, naturally, everything fits with no space. On the second configuration, the two larger shapes swapped perfectly, but there was a "minor" problem with the smaller "L" shaped pieces; they didn't quite fit. 1/8" off - very close, but obviously not an exact fit.
As was shown in one of those illustrations in the links, there are little, long, narrow strips of "tolerance" there (shown in red). First, realize that there MUST be some tolerance there for this to work. Second, realize that the tolerance must be hidden.
On my model, I did this several ways. On some of the edges, I sort of rolled the edges over with a pencil. I did trim off the "L's" about a very thin toothpick's worth. Finally, I widened the gap between all pieces by running my pencil down in the groove, which also darkened it and hid any differences.
Now, it works just like the one in the video; it makes people scratch their heads (done made my neighbor scratch his! hehe).
BTW, the explanation where he said the diagonal line was not straight? Wrong. That does not account for the missing square (these are called "Missing Square Puzzles"). Mine was dead straight.