In many chemical reactions, electrons move. For example, if we take zinc metal and oxidise it, we convert it to a zinc salt, which contains zinc ions, which are positively charged.The normal zinc ion has two charges per atom, which we write Zn++. So each zinc atom has lost two electrons. They have to go somewhere. If the zinc was oxidised by “rusting” it in air, it would be the atmospheric oxygen that would take those two electrons, ie the O2 oxygen molecules are “reduced” to O-- ions. The result would be ZnO, zinc oxide, an ionic solid made up of positive zinc ions and negative oxygen ions. “Reduction” is the counterpart of “oxidation”, and oxidation doesn’t have to be done by oxygen, it can be done by any “oxidizing agent”. In essence, oxidation is giving up electrons, and reduction is accepting them. Whenever an oxidation takes place, giving up electrons, a reduction has to take place, accepting them, so that the total number of electrons remains constant.
The clever trick in an electric cell is to separate the oxidation and reduction reactions into separate locations – the anode and the cathode – so that electrons delivered from the oxidation (at the anode) flow to the reduction (at the cathode) through an electric circuit that connects them. Since we need a complete cycle, to make the total charge balance, we need to connect the electrodes by a second route, which is the electrolyte (a solution containing ions), which allows the charge to balance by the flow of the ions in that electrolyte. The lemon or potato, (or something more sophisticated if you look at the electrochemistry lessons I mentioned) acts as the electrolyte connection.
Having actually set up our two electrodes, the reason that the current actually flows, powering our electric gizmo, is that the oxidation and reduction reactions are, on balance, energetically favoured. The oxidation of zinc to zinc ions generates energy, so it "wants" to happen. (Actually it is a bit more complicated than just energy, because entropy comes into it too, but that is beyond the scope of this lesson.) In the case of Ronald’s lemon/potato cell, the reduction at the cathode is the reduction of hydrogen ions to molecular hydrogen. Now that is not energetically favoured, but the oxidation of the zinc is more energetically favoured than the reduction of the hydrogen. So the zinc oxidation is powering the reduction of the hydrogen ions, and has some energy left over to power the electric circuit. Or alternatively, some of the energy in oxidising the zinc is lost in generating the hydrogen. Of course, we could get some of that back by burning the hydrogen.
(You can probably see here why it was a simplification to say that the lemon/potato cell worked because copper is more electronegative than zinc. What actually matters is the potential of the Cu/H+ electrode in comparison to the potential of the Zn/Zn++ electrode. Likewise in the zinc carbon battery, the cathode is a C/MnO2 electrode, rather than just carbon.)
The original energy to power this potato/lemon cell came from the manufacturing process to make the zinc metal out of zinc ore. There are fraudulent adverts in various places which give the impression that the energy comes from the lemon or potato, so that this is a way of harvesting the sun’s energy in growing the plant to convert it to electricity. Ronald appears to have seen one such fraudulent advert in one of his earlier posts, because he suggests, as they do, that the energy came from the plant matter, not the metal.