Contrary to other answers, the zero potential energy of a circuit isn't the ground, which some would use to short circuit, but sum of all potential differences around any closed loop in a circuit is zero (it's a completely relative term, not at all like there is say in temperature).
It's a measurement of differences not a set scale of absolutes to some ideal molecular state by any standard. The "ground" might be a potential building lightning strike.
No, here in electrostatics, and circuits potential is entirely in the difference of unequally distributed electrons. If there was a statically charged anode or cathode there is a potential difference between either and the ground. The Ground (as in grounded to the Earth) is simply a way to diffuse electrons, but I use the earth as a cathode in my electric fence. So it's all a relative scale, not an absolute one.
An elecro-potential difference is commonly called voltage, but have to have such in context. The phrase you've quoted is only applicable to a particular situation. The Energy of a field is more like the force applicable on a charge in that area. The Voltage Difference is the amount of inequality of the charges (number of electrons) that are on one side of a plate that are not on the other opposite plate (anode and cathode). I simply happened to have recognized the quote.
The potential here is how much it takes those charges to equalize or stabilize or have zero difference, else there is a constant force where those electrons wish to jump across that barrier or may even travel backwards through a circuit. In nature electricity is always in flux, lightning might spark, heat from impact or radiation, ever changing, here it's the repulsive/attractive force of in defined model, based on voltage between two parallel wires or plates (educated guess). The only way voltage is ever defined is by the Potential Difference, requiring comparison to another, it's not like a ball at some height or even particularly gravity or even temperature. It's a measurement of inequality, one having more or less than another and the distance between them. Not only does one have electrons, but the other is missing electrons having waiting proton atoms/molecules which will normally attract an electron.
It's always better with a diagram with each and every question or answer, but in electrostatics it's the electron charge or lack there over that defines the energy being talked about, each of physics being compartmentalized so to speak while the "Energy" being something prolific throughout and it looks like similarities in working certain parts of it.
In electrostatics, it's only talking about the positive or negative charges, which would fall off with the distance from the charge normally, but by observation, we can tell that a relative "close" distance between and anode and a cathode that the potential created by the difference of the charges in the plates:
^ h v
Is constant between the two CLOSE plates, homogenous so to speak as the distance from the plate that pushes increases, the plate the attracts decreases maintaining a homogenous field and vice versa, so any change based the height between two such equally but oppositely charged equal length plates is constant. Basically whatever was based on height cancels out, all else being equal.
If you studied perhaps the gravity field in a perfect hollow sphere where gravity is zero everywhere which is also based on a premise of location, not a particularity of energy or gravity.
The statement is the EQUATION of the field. The E = (V-IR)/L where the field is INVERSE of the Lenght. Where the same amount of charge or voltage spread over a long length or areas would diffuse the electric field.
This sort of abstract common sense really only comes with years of physics, when it's a person begins to think on the principles of understanding rather than memorizing facts, once you've seen the same types of thinking a huge variety of different scenarios from gravity to electronics it starts to synthesize. (I wrote a book on this, once you know the principles then a person can then contribute to furthering understanding and come closer to the great thinkers who've been immortalized.
Coulomb or Ohm names are forever known, but I'd like to know first realized this little mental math limerick. But know you now what principle this little question is based on, you now have to write it in an equation that shows the height cancels out of the equations I imagine you were given.