starting from the left side:
The BC547 is configured to oscillate at the crystal frequency. The 47K ohm base resistor and 680 ohm emitter resistor bias the transistor to a reasonable Qpoint so that it can provide current gain. The voltage "generated" at the crystal is amplified by the transistor. Emitter voltage is fed back to the base through the 39pf capacitor, scaled by the 8.2 pF capacitor resulting in a net positive feedback at high frequency. The feedback is most effective at the resonant frequency of the crystal since the crystal presents a high impedance at that frequency. Once the oscillator begins oscillating, operation is sustained by this positive feedback.
Some of the emitter current at the crystal frequency is fed through the 10 nF capacitor to the BC337 which is configured in common emitter to provide current gain. The 470K resistor feeds DC bias to the base and raises the base voltage up to about 0.7 volts. The gain of the transistor insures that the emitter current is about 1 mA in this configuration. The high frequency signal applied to the base varies the current at the emitter and so also at the collector, and therefore varies the voltage at the collector. L2 and the 39 pF capacitor are tuned to form a resonant tank circuit that offers a relatively high impedance at only the crystal frequency, so the amount of voltage at the collector at this frequency is maximised. The antenna is connected directly to the collector of the BC337 and so has the collector voltage applied to its input. As a result, it accepts some RF power from the amplifier and radiates that into space.
Although it is not entirely clear, the NE555 is probably used to turn the oscillator and amplifier on and off by applying pulses of 9V to their power supply rail.