Why do half filled and completely filled orbitals have extra stability?

The stability of half filled and fully filled orbital depends upon to factors:

 Symmetry: The half-filled and fully-filled orbitals are more symmetrical than any other configuration and symmetry leads to greater stability.

 Exchange Energy: The electrons present in the different orbitals of the same sub-shell can exchange their positions. Each such exchange leads to the decrease in energy known as Exchange Energy. Greater the number of exchanges, greater the exchange energy and hence greater the stability. As the number of exchanges that take place in the half-filled and fully-filled orbitals is maximum, thus exchange energy is maximum and hence maximum stability.