The centre of rotation of a point on the smaller circle is the centre of the smaller circle. Therefore for a full rotation the smaller circle will travel a distance of its circumference around the larger circle.
But the centre of rotation of the centre of the smaller circle is the centre of the larger circle. Hence it rotates around the circle by the larger radius as suggested.

The centre of rotation of a point on the smaller circle is the centre of the smaller circle. Therefore for a full rotation the smaller circle will travel a distance of its circumference around the larger circle.

But the centre of rotation of the centre of the smaller circle is the centre of the larger circle. Hence it rotates around the circle by the larger radius as suggested.