The general consensus is that the (fake) cooling effect (i.e. menthol does not physically cool the skin, e.g. through evaporation, but interacts biochemically with the temperature receptors in the skin) numbs the skin, so that you don't feel the rash and/or nicks from shaving.
Rather like this former colleague, who used tiger balm (is that an acknowledged product identifier in the UK -- it is one of these chinese muscle ointments that create a hot sensation on the skin, primarily due to the presence of oil of wintergreen) as a headache medicin: he rubbed it on his forehead whenever he had a headache, since, as he said, the burning sensation was stronger than the original headache...