The analogy is not with juice as in orange juice, as suggested by Dobhran Black’s answer to Could someone tell how electric power resembles juice?. Clearly there’s an analogy with fluids to be made; but why juice and not water? Or quicksilver?
The analogy is with vital juices, a concept that was kicking around as a literal concept from the ancient Greeks until modern medicine, and that indeed persists even now, both metaphorically, and in reference to plants (to judge from Google Books).
Vital juices encompasses the fluids moving within a living organism, that allow it to keep living. If the vital juices are flowing through the organism, then it does things like grow (if it’s a plant) and move (if it’s an animal).
If there’s electricity flowing through a machine, then it does things that resemble life: it is animated, so to speak. It moves, it whirrs, it does things.
So, the similarity is both the fluidity of electricity, and the vitality it confers.