It has to do with the time constant of the realistic model of a capacitor. This realistic model of a capacitor is the model of the parasitic complex impedances which are present in electronic devices.
The model for a capacitor looks like this:
R_ESR is the equivalent series resistance of the capacitor.
L_ESL is the equivalent inductance.
R_Leakage is the leakage resistance.
L_ESL is the inductance due to the leads on the capacitor. They may not be very long, very thick, or look like what you might believe to be a capacitor, but they do in fact have capacitance. In certain applications this is very important. Generally speaking, this is why you want the leads of your capacitors short and thick, and you want to place them as close to the component needing the capacitor as possible. The reactance of the capacitor is cancelled out by L_ESL, so for small value capacitors for instance, this plays a significant role.
R_Leakage is due to the inherent loss of the capacitor material. Put it this way, a capacitor is not a perfect device, it leaks. A capacitor will in fact pass a small DC current when placed in series in a DC circuit, but for the most part this is negligible.
R_ESR is the equivalent resistance of the capacitor of all elements in series with the capacitor. The leads will have a small resistance, due to their finite width. As current flows in a circuit, resistance will impede it's flow. No material is a perfect conductor, and will have a certain per unit resistance. Additionally, one can take into account the resistance of the connection between the leads and the capacitor plates.
Now, back to the time constant. For a simple RC circuit, the time constant Tau is R multiplied by C, where R is the resistance in series with the capacitor, and C is the value of the capacitor. A large value of Tau means the circuit is slower to charge and discharge the stored energy in the capacitor.
Now take the realistic model of a capacitor: What do you have? That's right, a very simple RC circuit! So a capacitor with a lower R_ESR will have a faster response to a change in the voltage: It will charge and discharge into the circuit faster than a capacitor with a larger R_ESR.