I would guess that one plate is being fed with the AC voltage. The AC might be tapped of the input AC or it is possible that a couple of the gates are being used to make an oscillator to feed the primary plate. When water is present then its resistivity is allowing some voltage to develop on the secondary plate which is then being detected by the schmitt trigger logic.
It looks like there are some other diodes so it could be the AC voltage from the secondary plate is being rectified to form a dc level that is then detected.
First you want to find the 0V (logic ground). I think this is the lowest thick trace on the board. You can check by measuring the voltage across the black component to the left of the 4 diodes near the 9V legend. It should measure about 12V dc and the polarity will tell you if the bottom is the 0V. It looks like it might connect round to pin 7 of the logic chip which would also be right.
You then have various strategies. One would be to measure the AC on the two plates with respect to 0V. One should be an ac voltage there all the time (primary plate). The other will only have an ac voltage when water is present. This is the secondary plate and would provide a signal that you could detect.
The second strategy may be easier. I would guess that one of the logic outputs from the quad gate (pin 3,4,10,11) is actually a water present logic signal. You could check these in turn and see if one of them just changes state (high to low, or low to high) when water is present. If so then you could use that directly via a resistive divider into a GPIO on the ESP8266 where the 0V of the ESP8266 is common to the 0V of this circuit.