When the gap between the probes is dry there is a near infinite resistance from the collector (and therefore the negative supply) to base of the 2N4403 transistor.
This transistor is connected to its emitter by a 100K resistor, but no current can flow so there is no voltage drop across this resistor.
So both ends of this resistor are at the same voltage.
So the base of the 2N4403 is at the same voltage as its emitter.
So the 2N4403 is off.
By similar reasoning the 2N4401 is also off since there is very high resistance from its collector supply to its base.
So the 1K resistor to its emitter also passes no current etc.
When the gap between the probes is wetted this introduces a modest resistance between the probes allowing current to flow from the battery negative through the wet probe resistance, the 1K resistor and the 100K resistor.
This sets up voltages to bias the 2N4403 on, so that the more the water, the lower the probe resistance and the greater the part of the battery voltage that appears across the 100K resistor. ie the wetter the probe the harder the transistor turns on.
Similarly when the 2N4403 turns on it passes collector current through the 220 ohma and 1K resistors, thus biasing the 2N4401 on and sounding the buzzer.
Both transistors contribute to the amplification of the small current passed through the water by the battery.
This is much easier to achieve with alternate pnp/npn (or npn/pnp) stages.
Incidentally it is much easier for all concerned to talk about a circuit if you label the resistors R1, R2 etc and the Transistors TR1, TR2 and so on, as well as giving values.
