Why the cutaway at the very beginning when he takes the first two chunks out? There's no reason for it, unless there is.
No sleight of hand. It's built in to the puzzle.
Due to the angle of the slice, he's combining (for instance) 1/4 square to a 2/3 square and to an observer it's close enough to be 1 but it is not.
The wooden frame is placed off-camera. Actually, there are two wooden frames of slightly different dimensions. The squares are not all square and are of slightly different sizes (see James above). All in all, a VERY ingenious and complicated presentation.
I'v had this vieo for years. I just watched it video 4 more times, and I cannot see any sleight of hand. Looking closely, the frame appears to be the same. But if it isn't... how could that matter?
I could see the squares being a slightly different dimension, and thus making the whole thing ± one square. But he does it with 3 squares!
All I can figure is, it's magic. Now if this guy can get lead to turn into gold...
[Here's another version of the same illusion: http://www.michaelbach.de/ot/ang-trianglePuzzle/index.html ]
Look at 2:35 and it becomes obvious.
OK, I've looked at about seven or eight YouTube vids of Curry's Paradox, including all the Tanton ones. I think I understand it now...
If you look at the squares originally in the frame, they are all perfect squares, all the same size. Then when he pulls them out of the frame there is a cut in the video and when he sets it up on the table the squares of the puzzle are not all perfect and do not line up quite right. So obviously this is a different set up which he then deconstructs down to the original set he showed you at the beginning of the video, which will once again fit inside the frame. It's a trick. A pretty clever one though.