"we don't need much voltage to separate the charges and therefore the
membrane capacitance is quite high"
I don't think this sentence is particularly useful. The capacitance of the cell capacitor, which is formed by two conductors (eletrolytes inside and outside the cell) and a dielectric (membrane), is determined by its physical characteristics, such as the thickness and the dielectric constant of the membrane and the area of the "plates", not by the voltage needed to separate the charges.
- What does it mean for a cell to use voltage to separate charges?
It is rather the other way around: the cell, or more specifically, the ion pumps built into the cell membrane, acting as a battery, separate charges, by shuttling ions across the membrane, and, by doing so, charge the cell capacitor to a particular voltage.
- What are the implications of a cell membrane being thicker?
A thicker membrane implies a smaller cell capacitance, which is similar to the effect of the dielectric thickness on the capacitance of a man-made capacitor.
...why would larger distances require a greater voltage?
Again, this phrasing is not particularly useful. To function properly, the cell capacitor has to be charged to a particular voltage level (a resting membrane potential), therefore, regardless of the distance (the thickness of the membrane), the ion pumps will continue moving the charges across the membrane until this voltage level is reached.