Yes, I got it.

i happens due to the displacement of 1/7 of a block from 7 different blocks in order to eliminate one. the way it is cut in across three rows of blocks makes it possible. Watch it slow and notice this. When he slides the upper part over to the side before rotating the blocks back around, look at how the block that just got moved from the first row is just slightly larger than the block that got moved back into that row. also notice that the separation lines are not perfectly straight anymore as he is working through the excercise. The reason that it still appears to fit in the box is that overall, there has only been 4.7% of the area removed, almost negligible. There are very slight differences that are almost un-noticable in the video but they definately occur.

Yea what he said

The trick is in the shapes of the pieces and in the cut pieces in the middle rows. First, they look like squares but some are rectangles with black painted edges of different thicknesses to make them look like squares. Next, when he slides the parts that have cut pieces to the sides to make new pieces he is really creating slightly shorter rectangles. The space gets made up by rotating pieces in the corner near him and on the edge near him. The thickness of the black lines bordering the pieces only makes them look like squares but they are really rectangles. If they were perfect squares, with all borders the same thickness, and not rectangles he couldn't do it without it being obvious.