Originally Answered: Vibrating plates & belts for fat burning, how does it work?
I've actually never seen any of these companies attempt to explain why their products supposedly work. They typical just list the "benefits" and state that they are "scientifically proven". Typical weight-loss industry horse ****.
Now for the real science. There was a review article in one of the bigger peer-reviewed journals (I can't seem to find it), that basically shot it down as a viable form of cardiovascular exercise. If I recall correctly, it burned approximately half as many calories per hour as cycling (not sure what the intensity of the cycling was). Several studies have shown a beneficial effect for the frail elderly, and it is being used in some places as a supplemental therapy for sarcopenia (severe muscle loss) and osteoporosis.
I absolutely do not think the (minor) benefits are worth the risk though. Have you ever heard of Hand-Arm Vibration Syndrome aka White Finger? It's caused by exposure to vibrating machinery. Some factory and construction workers get it and it is not pleasant. Most of the scientific publications will also point out the risks of organ damage, intervertebral disc damage, and various neuropathies. Again, not pleasant.
EDIT: I forgot to mention the theory of how it stimulates muscles activity. It's basically a neuromuscular effect. It's thought that the vibrations activate minute reflex activity in the muscles, causing them to tense. It is probably related to the tonic vibration reflex, though I only know that term in relation to Hand-Arm Syndrome, which I mention above, and cerbral palsy. It's sort of related to spasticity. Maybe start by looking into tonic vibration reflex. As for fat tissue, the vibration should have no effect.