A wonderful explanation of why is rotation of , where is most easily interpreted with the complex-number like structure of with normalized (i.e., ). Note that is then just (since just like the imaginary for complex number).

Basically quarternion is 4D stereographic projection back to 3D space. When multiply by a quarternion from the left, points parallel to will be translate along $\bf v$ and points penpendicular to will rotate around (following the right hand rule). Similarly, when multiplying from the right by the same quarternion, points parallel to will be translated in the same direction, but points perpendicular to will be rotated in the opposite direction (following left hand rule). So if we multiple and from both left and right hand sides, the translation effect will be exactly cancelled, and the the rotation will be doubled. That is why the angle in should be halved.