In a constant pressure process, part of the heat supplied gets utilized in increasing the volume of the system, or in other words for expansion. Whereas in a constant volume process, all the heat supplied is used in increasing the internal energy of the system. Internal energy being a function of temperature, the temperature of the system rises. The volume of the system being constrained, there is no scope for expansion.
Hence more amount of energy has to be supplied to a constant pressure process to increase the systems temperature by the same amount, because some of this energy is lost due to expansion work done by the system. Hence the specific heat at constant pressure always exceeds the specific heat at constant volume.