Mechanism of transformer secondary current
This is the way it was taught to me and is, in my mind, fairly easy to understand.

Consider the transformer circuit below.



A magnetic field is generated by the source. That field effectively creates a back EMF across the primary. In an ideal transformer, with no load on the output, when the field is string enough that back EMF cancels the applied voltage so no current can flow.

When a load is applied, the current taken by the load tries to change the field in the transformer. The change in field in turn reduces the back EMF on the primary. There in now an imbalance in voltages on the primary side, so current will flow from the source to sustain the transformer field.

The ratio of how much the load current tries to change the field to how much the supply needs to apply to fight it is governed by the turns ratio. The energy expended on both sides being equal. 

In many ways, a transformer is a true flux capacitor.

Of course all this happens at light speed and sub-atomically and the net macroscopic effect on the transformer flux is zero measurable change.

EDIT: I see a few negative responses to this answer which I am sure are being generated either by my poor writing or by stoic adherence to Savart, Maxwell and Faraday's definitions. That is fine. However I'd like to point out that the latter equations and theories relate to the macroscopic measurable effects not the quantum and time dependent effects at the atomic level. There is interesting literature out there on Perturbation Theory and in particular Reconstruction of Macroscopic Maxwell Equations: A Single Susceptibility Theory

EDIT 2:

Since I don't think I explained my model very well I'd like to show you it in a different way. 

For the moment, lefts take flux out of the picture. This example breaks the transformer into two ideal and equal motor/generators locked together by gears as shown below.



The coils in the primary now spins a rotor, which in turn spins an equivalent rotor in the generator. The speed of the motor is governed by the Back EMF Constant, and in an ideal motor will self regulate to that voltage. 

As you can see, the output voltage follows the same formula as that of a transformer that is, \$V_{Secondary} = V_{Primary} * N\$.

When current is drawn from the secondary it applies a braking torque to the generator. In order to maintain the regulated speed, the motor MUST apply the same torque to balance that braking torque. The motor must therefore consume the appropriate amount of current to generate that torque.

Again the formula for that current transfer follows the traditional transformer equation of \$I_{Primary} = I_{Secondary} / N\$.

As such you can see the above model accurately represents how we understand a transformer to work.

Of course a transformer does not quite work that way since the above implies it would work in DC. This model really needs to be extended to make the primary motor an ideal rotary voice coil where it's Back EMF Constant is measured in Volts Per Degree of Rotation, however the math still works out the same.

In an actual transformer the same thing is happening, only the motions are taking place at the atomic level as electrons and atoms re-arrange themselves in the dance that is electromagnetism.