How does capacitive EEG electrodes measures EEG signals
The key element here is that they are measuring charge displacement.  In a similar way than a high pass filter where when a signal is moving on one plate of the capacitance of your C-R filter and can be picked up on the resistor side.

You can also think of as that electrode acting like an antenna picking the signal emitted from the brain.

In our case, several considerations here:


You want to have the cutoff frequency of your 'C - R filter' electrode system to be as low as possible since the typical brain activity frequency - which I am supposing is ~few Hz. Which means large C and large R.
The large resistance is usually obtain using op-amp, instrumentation operational amplifier in particular.
The C is given by the capacitance between your brain and the electrode. Your body tissues (bone, skin, hair) act as dielectric here just as in a regular capacitor.
Typical value of coupling between the electrode and the brain can be very low especially if you have a lot of hair, keep in mind that the value of the capacitor is to first order 1/(distance to brain), so you want to make that distance as small as possible.
A bigger plate of metal will also give you a higher capacitance.
However a bigger plate will also average brain activity over a bigger region.
The signal from the brain activity is quite small, therefore low noise, high gain electronics is required amplify the signal from the electrode.
Because the signal from the brain is so small your electronics has to be sensitive to very small charge displacement is therefore very sensitive to pick-up from any other electromagnetic field in the vicinity (probably a good idea to have a notch filter in your readout electronics). The instrumentation amplifier helps here too (for common mode rejection in particular).