Take two capacitors that are each connected to and charged from identical constant voltage sources. The capacitors are identical in every way (distance, area, etc) except that their dielectric constants are different.
Capacitance is proportional to the dielectric constant, so all else being equal, the capacitance of the cap with higher dielectric will be greater.
According to the equation $Q = CV$, if the capacitors are both charged to (and left connected to) the same voltage then the capacitor with greater capacitance will hold more charge.
My question, assuming I have understood correctly, is why? What is physically happening on the cap with the higher dielectric that allows it to hold more charge on its plates for a given voltage? I am seeking an intuitive understanding.