Why half-filled and fully-filled electronic configuration are more stable?

The exactly half-filled and fully filled orbitals have greater stability than other configurations.

The reason for their stability are symmetry and exchange energy.

(a). Symmetry

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

(b). 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.