Ignore (for now) the MPSA18 and the two zener diodes and 10M resistor - they are used to control the amplitude of the DC output once the circuit is producing high voltage: -
Upon power being applied the 10uF capacitor has a charging voltage (in red) rising from 0V and after a short while this voltage will cause the base-emitter of the 2N4403 to conduct which rapidly causes the MPSA42 to switch on via the 1k8 resistor.
The MPSA42 will switch on and immediately start dischrging the 10uF capacitor via the 1k and the 1N4007. Shortly afterwards the MPSA42 will turn off because the 2N4403 turns off due to the cap discharging.
The current that was flowing in the transformer primary has stored energy in its magnetic field and this is harvested by the secondary circuit which presumably has a higher turns ratio than the primary.
And the process starts again - MPSA42 turns on - energy is stored in magnetic field and discharges to sencondary when MPSA42 switches off.
The MPSA18 will start to conduct when approximately 240V is on the output and this will start to turn-off the 2N4403 more making the 10uF take longer to charge thus the duty cycle is reduced.
It looks to me like there will be a fixed period of a few microseconds during which the MPSA42 will conduct and an ever-increasing period where it is switched off as the output DC level gets to about 240V DC. That makes sense.