Generally, it snows more near the freezing point than at temperatures further below freezing. Here's why.
Snowfall (like all precipitation) is dependent on something called "relative humidity", which is a ratio of the partial pressure of the water vapor in a mass of air compared to the saturated vapor pressure (i.e. the water vapor pressure at equilibrium with any liquid water present). 100% relative humidity at a given temperature is also called the dew point, and an air mass at 100% relative humidity is known by the arcane technical term of "cloud" (or if it's close to the ground, "fog"). The dew point varies with temperature, however: 10 grams of water vapor per kilogram of air at 15 degrees C would be almost 100% relative humidity, but that same partial pressure at 25 degrees C is only about 50% relative humidity. When a cloud is cooled for some reason (whether that be due to a front, the sun setting, or even the cloud being pushed up a mountain slope or just rising due to the adiabatic process), it suddenly becomes super-saturated (i.e. the relative humidity increases to more than 100%) and some of that water vapor will condense to bring the air mass back to equilibrium. If the cloud is colder than the freezing point the moisture comes out as snow. But because a warmer cloud can hold more moisture than a colder cloud, the closer the cloud is to the freezing point the more snow it'll produce.
Note that this doesn't necessarily mean that the ground is near freezing; it can be colder on the ground than it is aloft. But normally those heavy, thick layers of big snowflakes that are associated in media with "cold" only pile up when the temperature is close to freezing. When it's colder, you get needle-like granular snow that doesn't stick together well enough to make snowballs and tends to blow around in the slightest wind, and you also get a lot less of it. Some parts of the Canadian Arctic only get three or four inches of snow a year.
The trope probably exists because most English-speaking writers live either in California or in southern England, both comparatively mild areas that rarely get snow. To people in these areas any time when it snows is "very cold", so they don't differentiate between 32F/0C "cold" and -40F/C actual cold. Writers based in New York City may be used to snow, but it will usually also be of the "big fluffy flakes" type. Some of these writers might not even know of the existence of granular or "dry" snow. Needless to say, the trope is usually averted in Canadian shows, since even writers living in Vancouver (where it rarely snows, and when it does it's the big fluffy flakes type) usually come from somewhere else, like Saskatchewan.
