Shared electrons contribute to the stability of molecules by forming chemical bonds between atoms. When atoms share electrons in covalent bonds, they achieve a more stable electron configuration, typically resembling that of noble gases. This sharing allows atoms to satisfy their valency and attain a lower energy state, increasing the overall stability of the molecule. Additionally, shared electrons create electrostatic attractions that hold atoms together, preventing them from separating easily. As a result, molecules maintain their structure and integrity, enabling them to exist in a stable state under various conditions and resist decomposition or chemical breakdown.