There are Maxwell's laws of electromagnetism which tell you that the rotation of the electric field is given by the time derivative of the magnetic field, and it is the rotation of the electic field which will give you a non-zero electromotive force in a closed conductor circuit and hence induce a current.
But that's nothing else, finally, but the mathematical statement that a time-varying field induces a current. So it doesn't give a *reason* for it, it just *states* that this is so.
So then the question becomes: why are the Maxwell equations the way they are ? Now, one can think of a few "answers" to that one (Abelian gauge theory for instance), but again, that's just, again, reformulating the statement in other terms. In the end, it is just, well, because.
Since no known field influence moves faster than the speed of light, there will always be a time variance. Even if a theoretical field expressed at 1000x the speed of light, there would be a time variance.
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