EMI is described by 2 laws, Ampere & Faraday. AL relates the flux with the currents, & FL relates the flux with the voltages. In addition, the law of Lenz plays an important role, as well as conservation of energy & Ohm's law, which are universal.
These laws are axioms, meaning that they are not derived from anything more fundamental. Lorentz later published his force equations which describe the force acting on an individual charge in the presence of E & B fields.
We know that e/m induction takes place, & we can predict its behavior mathematically, but I don't think anyone can say WHY. It's like asking why gravity exists, or why electrons & protons have charge while neutrons don't. It just happens to be that way. Sorry if that is not the answer you are looking for, but that is quite a tough question.
Claude