You hear this expression a lot, it was -6 this morning and a friendly elderly chap quoted that exact sentence. Who knows the science behind this statement - if true in the first place??
Probably because the absence of clouds makes it colder as there is nothing to contain the heat rising from the earth. If there are no clouds you can't have snow (or rain).
It depends on other factors. Cold still air is dense as well as dry and precipitation usually results from rising air, but the cold air you feel at the earth's surface may have warmer air above it in which condensation is taking place.This could lead to snow hail sleet or rain according to the temperature of the air beneath. Like all generalisations about meteorology it contains a germ of truth but is mainly cobblers.