Explanation of this drawing (transistors)
This isn't a circuit I'd throw at a raw beginner!
Your picture of "current" vs. "voltage" is incomplete.  I'm going to re-describe the operation for you, trying for the terminology that'll give you a good mental picture of what's going on.


With no water present:

There is voltage across both transistors, but neither has any voltage on their bases, so the transistors remain off.

With water present

The water sensor becomes conductive, allowing current to flow through the 330k-ohm resistor into the base of the NPN transistor (the BC548).  Because the base of the PNP is at a higher voltage than the emitter of the NPN, this causes current to flow from the PNP base to the NPN emitter, which turns on the PNP.

Here it gets complicated.  When the PNP transistor turns on it pulls its collector voltage higher.  This raises the voltage on one end of the 10k resistor, which increases the base current of the NPN.  At this point each transistor is acting to turn the other one on harder: this condition is known as positive feedback.

This state cannot last, because the capacitor starts charging up and the current into the base of the NPN drops.  At some point the NPN's base current drops to the point where the PNP's base current starts dropping.  At that point the PNP's collector current falls, and the speaker's voltage drops.  When that happens the capacitor and 10k resistor try to pull current out of the NPN's base, and it turns off more.  You have the same positive feedback situation, only it's acting to turn off both transistors.  Eventually the capacitor discharges completely, the NPN starts turning on gently, and the process repeats.

If the components are sized right, the whole cycle takes between 1/200 and 1/2000 seconds to repeat, and you hear a tone in the speaker.  The frequency of the tone will be dependent on the resistance of the water sensor, the temperature, the transistors chosen, and a whole slew of other things you cannot control.