The combination of cold, dry air outside and over-heated dry air inside leave your skin feeling dry and chapped.
As the temperature drops outside, the air is less able to hold moisture and so it becomes drier. This prompts it to draw moisture from your skin, leading to that dry, tight feeling.
On top of that indoor heating, wood-burning stoves, and cosy fireplaces all reduce the humidity indoors and dry your skin out even further.