It doesn't sound like you want the electric/electronic explanation, but instead more of a physical/physics explanation. So I'll focus on that.
Think of your secondary as a conductor of some given length. Don't think about the primary, yet. Don't think about the varying magnetic field nearby it. Just think of the conductor. It's a veritable sea of electrons floating around in the "conduction band" due to thermal agitation. That's what a conductor IS.
If it helps any (and it's not important for thinking about this), the number of electrons is in an equilibrium state which merely means that an equal number of electrons are entering the conduction band (due to thermal agitation) as leaving it (due to recombination.) So it's not a static "sea" but it is a "stable" one.
These electrons repel each other. That's a given. So what takes place is that they get as far from each other as possible within the volume of the wire. So this means they are uniformly distributed, except at the very surface, where a few more can accumulate to create a very slight static charge (not many -- we are taking about ones and tens, not millions or billions by any stretch.) So the basic idea here is that the density is essentially uniform throughout for all practical purposes.
Now add to this an impressed external magnetic field that is changing. This magnetic field generates a non-Coulomb electric force that will act on the electrons in your secondary wire.
But this is like putting a weight onto the floor, pressing down on that floor due to gravity. The floor doesn't really move much. It might flex a little for a moment. But quite quickly counter-acting forces act to counter the added force and no more work is done. No matter how long the weight sits on the floor, no work is performed or required. It just sits there.
Similarly, these non-Coulomb forces could accelerate the conduction band electrons into motion.... if there was somewhere for them to go. But there isn't. You haven't connected anything up to it. So all that happens with these varying non-Coulomb forces is to press this way, then that way, then another way, on these conduction band electrons. They might move around just a little bit. But only a little. This is because as soon as one tries to move, it immediately gets repelled still more by some neighboring conduction band electron (itself trying to move, perhaps.) And because electric repulsion forces are so strong (about 40 orders of magnitude greater than gravity), in almost no time at all that "sea" has "flexed slightly" in response and ceases to move more. After all, the ends of the wires can only accumulate a few extra electrons (at most) before the repulsion is enough to completely stop any additional motions in the wire.
So we are talking about a few (one or two or three) electrons accumulating on one end of the coil's end, then the same few on the other end, then back again. There might be some slight changes at the wire surface in between. But the gist here is "nothing much changes or happens." It's just a slight jostling around.
The work done by these tiny motions is similarly tiny and probably not easily measurable. (I don't know of any standard COTS instrumentation capable of it, though I'm sure that physicists have from time to time played around, trying to.)
Now, if you provide a load of some kind, then this can allow those non-Coulomb forces to operate. And they can be easily powerful enough to induce sizable currents. (It only takes one or two electrons at a 90-degree corner in a large copper wire to cause an amp of current to make such a turn with ease -- such is the incredible power of electric forces.)