When the Darlington transistor turns on, current flows though the primary of the transformer to produce magnetic flux in the core.
This flux cuts the turns of the secondary and since there are many more turns of the secondary, the voltage produced by the secondary is many times greater than the voltage across the primary.
But in this circuit, the high voltage produced by the secondary at the moment is not as high as will be produced in the next part of the cycle and we will not be concerned with this fairly low voltage.
The output of the 555 is a square-wave and after the transistor has been turned on for a short period of time, it is abruptly turned OFF.
This is when an amazing thing happens.
The magnetic flux produced by the primary abruptly stops.
The energy in the core of the transformer (in the form of magnetic flux) ceases to receive magnetic energy. This magnetic flux is called expanding flux, even though it may not be received at an increasing rate at the time of the stoppage.
As soon as the receiving flux (energy) is turned off, the energy in the core starts to collapse and when it does, the magnetic lines of force reverse direction.
These magnetic lines of force cut all the turns in the transformer (both the primary and secondary) and the collapse is very rapid.
This produces a voltage in the turns that is not only very high but in the OPPOSITE DIRECTION.
The actual speed of collapse is called the “Q-factor” of the inductor (transformer) and this is also known as the “Quality Factor.” The Q-factor can be 10, 100 or even 1,000 and is the ratio of the voltage produced by the output when a sinewave is delivered to the transformer compared to the voltage produced when the input voltage collapses very quickly.
There is no way to determine the “Q” of the transformer from any winding data or size or shape. The transformer must be used in a circuit to determine this.
As a side point. The BD679 transistor has been tested to have a breakdown at about 270v - as proven im my Xenon circuit - so the 80v rating in the data sheet has no bearing.