Like John Dallman pointed out, living beings need to get both energy and matter from their food. That is true for any living being, including bacteria and the such. They don't need it to come from a single source, but they need both.
Said energy needs to be in a form that they can process, usually chemical energy from molecules like glucose that can go through some chemical reactions to be converted into another molecule that can be used in many different metabolic processes (think ATP).
The matter portion is where they gain the base components of their bodies and can be used for all sorts of things, like structural uses (calcium being used on bones, hard chitin on insect's exoskeletons), substrates for chemical reactions (sugars like glucose for generating the energy, amino acids used to synthesize proteins which are then used in a sorts of ways), catalysts (things like vitamins and some ions), etc.
If you want it to make sense scientifically, you need to decide a source of matter for them (or just give an explanation that makes sense for your world in case they have things that are beyond science, like some kind of magic system).
Since you seem to intend for them to damage electronics, I would suggest you make them their source of matter. Something like a being that digests electronic components and uses them in their body. Maybe that is why they even use electricity on the first place. Maybe their main source of chemical energy isn't some oxidization of sugars in several steps (like most living beings), but instead electrolytic reactions (reactions that use an electrical current to occur).
Electrolytic reactions can be used to extract metals from ores, so I don't think it is that much of a stretch that some fictional creature could use electric currents on electronics and power lines as the energy they need to make a electrolytic reaction that converts portions of the metal in the cables or electronics into a form that they can absorb (it would be way more complicated than that, but as a concept I think it can pass). Something like that would also damage the electronic a lot too.
If you go with that, the creature would probably need a lot of metal to survive (as to justify said method of getting food), so maybe it uses that metal structurally (like metal bones or exoskeleton, although at this point they would start looking more like robots)? Metallic ions are good catalysts so that is an option, but catalysts are consumed in small quantities, so the creature wouldn't need much of them. Also, unless they are made only of materials found in electronics, they would likely need another source of matter for the other components in their bodies, even if said source is only needed in small amounts.
As for characteristics that they would need to manipulate the electricity with their bodies, I think they would need physiological structures for both conducting it really well (think materials with low resistance like metals) and conduct it really poorly (hydrocarbons are good at that, but the creature would need a way to get them) so it can control how the current flows through it, directing it to where it needs to go and avoiding it from going where it shouldn't.
Now, if my memory serves me right, electricity always will "prefer" to flow through the material of least resistance (they are inversely proportional), so a way of making anything electronic on that circuit useless while the creature eats could be by it having a material on its body whose resistance is so low in comparison, that the creature basically just takes all of the electricity instead.
Unfortunately, the amount of energy on cables these days is actually quite high, so the creature would either need to take a lot of energy and convert it with immense efficiency or it would dissipate a lot of it as heat and heat usually increases the resistance of materials pretty fast. Even if the creature can withstand it, the electronic and/or cables may not.
One way around it could be to have the creature have some kinda of room temperature superconductor on it (these don't currently exist as far as humans know, but there are superconductors at very low temperatures and a lot of research is being done on how to create one that has such properties at room temperature). If I understand it correctly, there would be no resistance to the flow of electricity on the creature, so it should not dissipate any energy it doesn't absorb as heat and, since the creature's resistance is so much lower than the one on the cables/electronics, the vast majority of electrical energy should flow to the creature first and I can see that creating a short circuity (I don't know how the electrolysis would be affected, or if it would be affected, by the use of superconductors though).