Shared electrons in covalent bonds contribute to molecular stability by allowing atoms to achieve a more favorable electron configuration. By sharing electrons, atoms can fill their valence shells, satisfying the octet rule for most elements. This stable configuration lowers the overall energy of the system, making the molecule more stable. Additionally, the formation of covalent bonds releases energy, further stabilizing the molecule. Overall, shared electrons enable the formation of strong bonds, leading to the stability of molecules.