The best intuition for me was the YouTube comment that said to imagine the rolling coin/circle is picked up on its edge, in 3D, rolling like a unicycle wheel. I

The best intuition for me was the YouTube comment that said to imagine the rolling coin/circle is picked up on its edge, in 3D, rolling like a unicycle wheel. It turns forwards to roll and measure distance, and it turns sideways to travel around the circular path. i.e. "N" rotations for circumference distance on one axis (the straight line distance) plus one full rotation on a different axis around the circle. When the coin is lying flat, both have to happen on the same axis.