In operation, the 100uF capacitor charges until the voltage at the junction of the 1M and 1M8 resistors is sufficient to turn on the BC557. This then turns on the ZTX869, which pulls the positive end of the 100uF capacitor down to ground. Positive feedback around the two transistors, aided by the 1uF capacitor and 4k7 resistor, ensures a sharp transition between states. With the positive end of the 100uF capacitor pulled down to ground, the sum of the battery voltage and the capacitor voltage appears across the LED and provides enough voltage to flash it. As the 100uF capacitor discharges, and the 2u2 capacitor charges, the BC557 runs out of base drive and the circuit reverts to the off state, again with positive feedback ensuring a sharp transition.
The value of 100k for the emitter resistor of the BC557 is not a mistake. The function of this resistor is not to supply emitter current, which is supplied by the 2u2 capacitor, but to discharge the 2u2 capacitor while the circuit is in the off state in readiness for the next cycle.
The ZTX869 transistor is a high-gain, low-saturation-voltage device from Zetex. It should not be replaced with a generic device. While it is possible to get something like a BC547 to work, its higher saturation voltage compared with the Zetex device will knacker the circuit's efficiency and it will only give a weedy flash.
The circuit is designed for red LEDs as the original "Pulse" LED is red. For yellow or green LEDs with their higher forward voltage, the 1M8 resistor should be increased to provide a higher trip point (though this may compromise performance on a nearly-flat battery), and it is also advantageous to increase the 100uF capacitor to 220uF and scale the 10k and 27k resistors accordingly. For blue and white LEDs it is necessary to cascade two output stages to obtain enough voltage.
