Yeah, it freaked me out, too. But this little cephalopod isn't actually alive - he's just
very freshly dead. A similar phenomenon can be seen in these frog legs:
How are these dead body parts being brought back to life?
These clever cooks are capitalizing on biology to put on a show. The key is that the muscle has to be fresh - very, very fresh. So fresh, in fact, that its neurons are still completely in tact and ready to fire.
All cells are polarized, which means the concentrations of charged atoms, called ions, of the fluid inside them is different from the fluid outside them. It is this difference in ion concentrations that creates a difference in charge (or voltage) across membrane. This difference - called a membrane potential - allows neurons to work.
Neurons are highly specialized cells which process and transmit electrical signals. When not activated, neurons maintain their membrane potential by actively pumping out sodium ions and pumping in potassium ions, keeping a voltage difference of roughly -70 milivolts. When a neuron is activated, however, specific channels open that allows sodium ions to flood in. This rapid change in the charge difference causes potassium channels to open, allowing potassium to flow out. At a certain point, the channels all close, and those active pumps work towards restoring the resting membrane potential - but not before triggering channels nearby to open. That's how the signal moves from one end of the neuron to the other - channels open in one area, which causes channels nearby to open, and so on and so forth from one end to the other:
When a creature dies, its neurons don't stop working right away. So long as there is still enough energy around to maintain that membrane potential, the neurons will work.
What you might have noticed is that in the case of the dancing dead, the cooks have added one key ingredient: salt (soy sauce is very salty). Salt - or sodium chloride - is chock full of sodium ions. This overwhelming dose of sodium ions is enough to trigger the still-working neurons into firing, signalling the muscles to contract.
So there you have it - take freshly killed muscle, add salt, and voila! - the dead shall rise again. Briefly, anyway.