A DC current generates a constant magnetic field with constant energy density. An electron coming into that field will be affected: an electron leaving that field will be affected. A secondary wire in that field will not be affected: energy change at one end where electrons leave will exactly balance energy change at the other end where electrons enter.
A moving electron is associated with a magnetic field (because relativity). Changing the speed or direction of the electron changes the energy in the magnetic field. There is no easy way to tap this magnetic field, but you can easily tap off changes in the magnetic field.
Accelerating an electron pumps more energy into the field (because relativity): changing magnetic fields accelerates electrons (because relativity) It will accelerate all the electrons in an adjacent wire: more energy in the electrons coming in, more energy in the electrons going out.
And because a changing magnetic field affects electrons, it will naturally find an energy balance where less energy is required, by moving two electrons in equal and opposite manner rather than pumping up the magnetic field (because thermodynamics: systems always find low-energy states)
You can pump energy into one side of a transformer, and energy will spill out the other side because that’s easier than energizing a magnetic field in the transformer core.
Real transformers do have to energize the transformer core at the best of times: we don’t get perfect lossless energy transfer. And if the current through the load would require more energy than pumping up the magnetic field, then the energy just goes into, and out of, the magnetic field, and the transformer is just an inductor with core and wire losses.