How does the step transition in this origami construction occur?
You have to pull out the paper as far as the vertical split visible in step 15. At the same time it will "wrap around" to show the other side of the paper.

You could also look at it like this: if in step 15 you completely unfold the white part, you will see a small square with a crease pattern such as for the bird base. Now, along the precreased lines, collapse one diagonal half of the square into a half of the bird base, and the other half of the square leave folded in two. This produces your step 16.