To supplement all the discussion of flyback action, this can be thought about in a more fundamental way.
The basic equation for an ideal inductor is: v = L di/dt
In other words, the voltage across the inductor is proportional to the inductance, and to the rate of change of current in the inductor.
Now, suppose you have a stable current flowing through the inductor and you suddenly switch off the current. In this case, the rate of change of current approaches infinity, since it has instantaneously gone from something to nothing. According to the above formula, the voltage across the coil will also approach infinite volts, since L is a positive constant.
In reality, the inductor/coil is not ideal, and so "infinite" can be replaced by "a very big number".
Secondly, once you have disconnected the current in the primary, the coil does not "know" which winding had current flowing before (it could have been the primary or the secondary--either way, there is a magnetic field established that has a store of energy).
Since the voltage produced across the coil will be proportional to the number of turns, the secondary side with more turns will experience a bigger "very high voltage" than the primary.