There are most likely three LEDs with different colors in that device which are driven by current pulses of different length and timing to achieve an arbitrary RGB color with the least amount of hardware. The result is that you are seeing a stroboscopic effect when you move your eyes fast enough to resolve the timing of the three colors. This is also known as persistence of vision, and the educational channel Veritasium covered it in this video, showing it in action.
If it's just a white LED with phosphors illuminated by a blue LED as in the answer by count_to_10, it could be the different phosphorescence times of the phosphors after a short pulse excitation by the one blue LED chip.
In either case the eye movement turns the temporal variation of the color into a spatial variation. While your eye is insensitive to the temporal variation (the designer of the device has chosen the "chopping" frequency just high enough to be invisible), it is quite sensitive to spatial variation. So what you are basically doing is to move different "frames" of the light to different locations on your retina.
