Common salt & water are neutral.Then how does distilled water become a good conductor when salt is added to it?

Water does not become a conductor ddependant on the acidity or alkalinity of the substance dissolved in it. It becomes a conductor because the solute contributes ions, which carry a charge, to the water. The fact that NaCl is neutral is immaterial - it contributes Na+ and Cl- ions, which make the water a conductor.