Suppose several different parallel-plate capacitors are charged up by a constant-voltage source. Thinking of the actual movement and position of the charges on an atomic level, why does it make sense that the capacitances are proportional to the surface areas of the plates? Why does it make sense that the capacitances are inverselyproportional to the distance between the plates?

Capacitance is the measure or the capacity of an electric device to store charge and allow the device to use the charge up as desired. The difference in potential causes the charge to get stored on the plates of conductor.

Mathematically,

$\mathbf{C}\mathbf{=}\frac{\mathbf{Q}}{\mathbf{V}}$

Where, Q is the total charge stored and V is the potential difference or the voltage applied electronically.

The capacitance is the capacity of storing or holding charge. When the area of plates is larger, the capacity to hold more charge is the possibility and this leads to a higher capacitance overall. The number of free electrons on the plates decides the measure of capacitance and thus if the electrons have enough space to arrange then the value of capacitance increases.

For the inter plate distance, the capacitance decreases with increase in the distance because the interaction between the plates drops.