Summary: There are two large right triangles, each composed of several smaller shapes. When the shapes are rearranged in the second large triangle, which appears to be the same overall size / shape, there’s an extra space.
I keep doing the math on the individual figures and the areas don’t add up to the correct area anyway - they add up to 32 squares, and the large triangle is 13 x 5 (using 3rd grade arithmetic, that’s 13x5 = 65, divided by 2, = 32.5).
The ratios also don’t make sense - for them to be able to move the triangles around as they do, they’d all have to have the same slope (ratio between vertic and horizontal axes) - that’d be the sine, trigonometrically speaking. I’m guessing that’s the key to the whole conundrum.
But it sure looks like the reorganization works! I just can’t quite get my head around how they did it and made it look correct.