Electric eels produce two levels of shocks. The low level shocks are used for navigation and prey sensing. The high level shocks can stun prey, but also hurt the eel a bit. The eel can also use the high level shocks for defense.
As Chronos noted, they aren’t actually using the shocks to power the Christmas lights, presumably because the amount of energy is small enough that the low voltage lights they managed to illuminate were rather underwhelming. Instead, they use the shocks to trigger other circuitry that causes the lights on the tree to go on. Basically, they are cheating to make the eels look more impressive at generating electricity than they really are.
In a way, the low level shocks are kinda like lemon batteries. Yes, they make electricity, but it’s really not practical to power anything from them. You’d need an entire table top of lemon batteries to power anything useful, and you’d need an awful lot of eels to generate a useful amount of electricity as well.
Also, lemon batteries, while weak, are at least constant. Electric eel discharges aren’t constant. They are very spotty and are of very short duration. Any practical electric eel power harvesting system would probably require storage capacitors or batteries of some sort that the eels would charge. A system like this would be theoretically possible, but would be very difficult to create in the real world. And, as noted upthread, you’d probably spend more energy maintaining the eels than you would get in useful electricity.
The high level shocks produce a lot more power, several hundred volts and about an amp or so of current (for comparison, the low level shocks are maybe half a dozen to a dozen volts, with a current measured in milliamps). But again, the eels don’t use this high level electrical discharge very often, and while the voltage and current levels are both pretty impressive, the duration of the shock is mere milliseconds. It’s more than enough to jolt the bejeezus out of you and get your attention, but the total energy is fairly small.
The shock organs in electric eels are basically weak chemical batteries. The reason they are so powerful is that there’s an awful lot of them stacked in series, which adds up the voltage between all of the individual electrical cells. If usable electricity is your end goal, there are definitely easier ways of constructing batteries and harnessing energy from them.
But, as a thought exercise, I think your biggest challenge will be keeping the eels alive. And you’re going to need a lot of eels. No, more than that. No, even more than that. I’m talking a LOT of eels.
Hamster wheels connected to mini-generators are probably a better choice.