Why does the current in a transformer's primary coil change when a load is attached to the secondary coil?
A more basic physics explanation is simply that conservation of energy holds*, and you can only stuff so much energy into the magnetic field of a transformer.
So if you're taking energy out of the transformer in the form of secondary current (and voltage), then you have to put energy into the transformer in the form of primary current (and voltage).  It's just inescapable, which is why perpetual motion machines are for con men and fools.
You can do an amazing amount of power electronics design just by remembering that conservation of energy holds, BTW.  Power in = power out.  Motors, generators, power supplies -- they all obey.
* On any human scale.  Shortly after you could build even a solar-system scale perpetual motion machine, you'd be ripped apart by runaway expansion of the universe.  But that would be after every proton in the universe has evaporated into photons.  So -- just take it that conservation of energy holds.