The reason for Cp greater than Cv is very simple.
This can be easily demonstrated by First law of thermodynamics.
At constant Pressure
At Constant Volume,
Therefore dQ=dU ---(2).
Therefore using the above relation Cp is greater than Cv.
Physical Meaning of why Cp is greater than Cv:
At constant volume, heat is required only to increase the internal energy of system while At constant Pressure heat is needed for increasing Internal energy as well as for doing the work to displace the piston of system through a distance. Therefore Cp has a greater value than Cv.
Hope it helps!!