I am watching this video over and over again and I can't visualize the current flow from the eel, to the two metal plates, to the Christmas tree. Is there a voltage difference between the two plates, thus causing a current to flow between the plate closest to the eel (highest potential) through the tree and back to the other plate (lowest potential), or does the electricity harvesting work differently?
The eel doesn't power the lights he only acts as the control signal. From an article about it;
Carnell and his colleagues installed two stainless steel electrodes, one on each side of Sparky’s tank. These electrodes collect the voltage the electric eel emits to then power a sequencer. “The sequencer takes the voltage the eel produces and operates circuitry that flashes the lights, fast or slow, based on the level of voltage he puts out,” says Terry Smith, project manager at Cache Valley Electric, in a press release.
The five-foot-tall tree, which stands just next to Sparky’s tank, is decorated with four strands of lights. While the eel does not power the lights, he does control the way the strands flicker. “As he shocks, one strand shuts off and another strand turns on,” says Allison.
Here's the full article
Edited to address questions
So to over simplify a bit most of the back part of an electrical eel is made up of tiny organic battery cells some in parallel and others in series down it's body. When it's ready to shock something, or just using it's power to look for fish, it fires some or all of those cells at once. It essentially becomes a large battery with a positive and a negative end. The current return path is spread out through the water at relatively close range. So if you're a fish in that path hey now current is flowing through you, but if you put electrodes at opposite ends of the tank now you've created another path for current to flow, and you can use that small amount of power to maybe move a meter, pulse a speaker like they do in that awesome 1950s youtube video referenced in the article, or design a system to control Christmas lights :)
Here's another link that talks more about the physiology of the fish itself.