During the electrolysis of aqueous NaCl, hydrogen gas (H$_2$) is produced at the cathode, while chlorine gas (Cl$_2$) is produced at the anode. This occurs because chlorine exhibits a greater oxidation potential than water, facilitating the oxidation of chloride ions to chlorine gas at the anode. Consequently, both the assertion and the reason are accurate, with the reason providing a valid explanation for the assertion.