It's actually kind of the other way around, it's not that being too cold prevents snow from falling, it's that an absence of moisture in the air allows the atmosphere to cool.
Low pressure systems bring with them (relatively) warm moist air which precipitates as snow. The clouds overhead act as a blanket, and keep the surface air warmer. When the low pressure systems, clouds, and warm moist air move on, it creates the perfect conditions for the temperatures to drop. Sometimes the warm air systems are replaced by cold air systems when the winds change directions. Where I live the temperature changes drastically when the winds change, on Saturday it was -20°C when the wind was out of the North. A Chinook has blown in the past couple nights out of the West (air from the Pacific Ocean that's traveled over the mountains and warmed up because of adiabatic lapse rates) and now it's +12°C. That's more than 30 degrees in only a couple days.
Cold air holds less moisture than warm air, but it isn't so much the temperature that prevents it from snowing, the temperature of the air determines how much moisture the air can hold. Cold temperatures are an indicator that the air has no more moisture in it, ergo there aren't any weather systems overhead that can produce snow.