Well, I have had a look at both your circuits and the signal one is a little complex, so I will explain that at a later date when I have worked it out myself! But the plant one I can explain now.
The circuit's main component is an op-amp. This is set up in a special configuration called a comparator. The IC compares the two I/Ps (pin 2 and 3), and then O/Ps the difference between them (pin 6).
The 10k and variable 100k resistors at the bottom of the circuit control the amount of feedback and amplification the op-amp has. In this circuit I would imagine the amplification is 1 or unity, ie: it doesn't actually amplify the O/P. The 3k3 resistor on the O/P is to limit the current to the meter, so it's not damaged. The two diodes on pin 3's I/P, are there to clamp the voltage going to the I/P so the comparator is as accurate as possible (I think!) The resistors on that part of the circuit are just there to keep the voltages and currents at controlable, safe levels. You wouldn't want too high a voltage going into your wet soil! The 1k resistor is there to protect pin 2's I/P and to try and match the I/P currents as much as possible to eachother. The +9v and -9v I/Ps at the top are to power the op-amp.
Now, for the function of the circuit. As current flows from one spike to the other through the soil, it will encouter resistance. This resistance will create a potential difference between the two spikes and it is this difference the op-amp uses as the voltages I/P to it will differ slightly. Depending on the dampness of the soil, will depend on the resistance in it. If the soil is very wet, it will have a very low resistance; if it's very dry, it's resistance will be high. This resistance is transfered through the circuit to the meter, which is an ameter (measures current). This can be calibrated however you want. So it would show very wet soil for high current (low resistance between the spikes) and a show very dry soil for a very low current.
I hope that explanation was clear and adequate enough. I will try and explain the signal circuit as well, at a later date.