He has skinned the frog legs and they are ready to be cooked. When he salts them, the muscles start twitching. The obvious question: why does that happen?

Because these are fresh frog legs, the cells inside them are all still intact. The biochemical machinery still functions. There is still a source of energy for the muscles in the form of unused ATP molecules stored in the cells. All that the muscles need is something to activate them and they can still contract and relax (until they run out of ATP or something else shuts down the biochemical machinery).