and W = -p dV = 0 along a line of constant volume.
So at constant volume, ALL added heat goes directly to the total energy. But at constant pressure, SOME portion of the added heat does work by increasing the volume of the system. So intuitively, the heat capacity at constant pressure is higher because the system is more "flexible" in the ways it can accommodate additional heat.
The precise relationship is more complex, but can be worked out mathematically.