Yeah, OP is actually correct that it's the field. If you have say a 700 kV power line, that means the voltage difference between the two lines is 700 kV. The laws of E&M mean that there has to be an electric field between the two lines, so if you took a charged particle from one line to the other, it would pick up a ton of energy. Incidentally, this is why working on high voltage lines is kind of intense, and the lines themselves have to be incredibly smooth (I think surface imperfections are micron scale or smaller to avoid coronal emission). It's the electric current moving through the field, both of which are provided by the generating station, which carry the energy. It's easy to forget that Maxwell's equations still apply to transmission lines, but they do!

To answer what I think is OP's question, classical magnetic fields don't do any work because the force is always perpendicular to the direction of motion, so the Earth's magnetic field doesn't do anything for power transmission. The Earth doesn't have a large scale electric field, because ions in the atmosphere would rapidly adjust to cancel it out. There aren't a lot of ions in the lower atmosphere (and again, transmission lines are very carefully designed to not create new ones), so the electric field doesn't get cancelled out and we can send power down the lines.