The lines either appear to curve gently or zig-zag sharply. Pretty much everyone perceives the lines that way, but that's the illusion. All the lines have the same curves. There are no zig-zags.
The illusion occurs because of the coloring of the lines, Takahashi explains. When colors meet at the tip of the peak or valley, it creates the illusion there's a sharp peak. The illusion is competing with the viewer's ability to see the curve of the line. This may be due to a conflict between the mechanisms that identify a gentle curve and an obtuse corner in the viewer's brain. One possible conclusion raised in the paper is when your brain is unsure which of those two it perceives, it defaults to understanding a bend as a corner.
“It is notable that observers exactly 'see' an illusory zigzag line against a physically wavy line, rather than have an impaired perception,” Takahashi writes. “We propose that the underlying mechanisms for the gentle curve perception and those of obtuse corner perception are competing with each other in an imbalanced way and the percepts of corner might be dominant in the visual system.”