Took me a while to follow that too! The three examples of fails at the top each show instances where there are non-consecutive parts of the snake on adjacent corners - it’s the lines highlighted in red.

Basically no two parts of the snake that aren’t directly joined to each other in the snake are allowed to be on corners which are only a line apart.

I think.

Any two coils that are not directly connected. For example, suppose we number from the head, coil 1, 2, 3, 4. Then pairs that are not directly connected are: (1, 3), (1, 4), (2, 4). The endpoints of these pairs of segments cannot be connected by an edge of the cube.