Consider a plane of charge. For distances from the plane which are small compared to the size of the plane the electric field will be constant. In the capacitor, with two oppositely-charged plates and relatively small gap, the electric field is a constant, depending only on the overlapping plate area and the amount of charge, $E_{gap}=Q/(k\epsilon_o A)$, where $k$ is the dielectric constant of whatever is in the gap.
For a constant electric field, the voltage between two planes perpendicular to the field depends linearly on the distance between the planes, $V_{gap}=E_{gap}d$.
The capacitance tells us the ratio of stored charge to resulting voltage, and is strictly a mechanical/geometrical result. The capacitance is independent of the actual charge or voltage on/across the device. If you change the area or the gap size, you change the capacitance.
It's similar to having a room. Depending on the number of people (charges) in the room, the people density (voltage) will change. If you change the dimensions of the room, you change the volume of the room, and the people density will change even if you don't change the number of people. While the actual dimensional meanings aren't the same, the idea that changing the mechanical properties changes the capacitance is the important point. Voltage and charge vary in accordance with the actual capacitance, so that their ratio is a constant.