All coronaviruses possess a fat layer as a component of the viral envelope. There is, however, no evidence yet that the SARS-CoV-2 (the virus that causes COVID-19) fat envelope in particular facilitates transmission at low temperatures.
This claim was likely derived from studies showing that SARS-CoV (the SARS virus from the early 2000s, not the one that causes COVID-19) exhibits structural stability at relatively low temperatures (22–25°C), though it was never shown that this was associated with it being "transmitted [between people or between surfaces and people] better in the cold."
But in laboratory experiments, SARS-CoV-2 does remain active longer in cold, dry conditions, especially when not in direct sunlight. And in general, cold weather is accompanied by dry air, and "turning on the heat dries both the air and the tissues lining the airways, impairing how well mucus removes debris and invaders like SARS-CoV-2." So it is important to continue to be vigilant through the winter - wear masks in public, distance, ensure good ventilation, avoid crowds, and maintain good hygiene.
Notably, enveloped viruses with a fat membrane are usually more sensitive to neutralizing agents (like chemical antiseptics) and conditions (like changes in temperature or pH) than non-enveloped viruses.
Updated 2 December 2020