If the plate with hbar written on it is rotated by 360o the strings cannot be unwound without rotating the plate, but with a 720o rotation the strings CAN be unwound without rotating the plate. Each point along the string represents some rotation, the two ends being the identity (no rotation). Therefore this device illustrates the fact that a one-parameter sequence of rotations beginning and ending with the identity cannot be continuously deformed to the identity if the total rotation angle is 360o, but can be so deformed if the total angle is 720o. (The rotation group is thus said to be "doubly connected".) This property plays a role in the quantum mechanics of an object with half-integer spin: the quantum state vector of such an object is brought back to its negative after a 360o rotation, and back to itself only after a 720o rotation.
A good description and animation can be found here: http://www.gregegan.net/APPLETS/21/21.html