I think it is possible to get a better picture by first thinking of a regular polygon, take a hexagon, for example. The hexagon has six corners, now if we initially started rotating a coin along the sides of the hexagon, once it reaches a corner, the coin rotates an additional 60 degrees ''around'' this corner, try and visualise this and hopefully it makes sense. Therefore, along with the rotation induced due to the coins motion around the sides of the hexagon, there is additional rotation along each of the six corners, and so one additional revolution is made once the coin reaches its starting position. Now if you think of a circle as the limit of regular polygons as number of sides tends to infinity, you can extend this logic to this problem to perhaps get a better intuitive sense as to why an additional revolution takes place.

I think it is possible to get a better picture by first thinking of a regular polygon, take a hexagon, for example. The hexagon has six corners, now if we initially started rotating a coin along the sides of the hexagon, once it reaches a corner, the coin rotates an additional 60 degrees ''around'' this corner, try and visualise this and hopefully it makes sense. Therefore, along with the rotation induced due to the coins motion around the sides of the hexagon, there is additional rotation along each of the six corners, and so one additional revolution is made once the coin reaches its starting position. Now if you think of a circle as the limit of regular polygons as number of sides tends to infinity, you can extend this logic to this problem to perhaps get a better intuitive sense as to why an additional revolution takes place.