As the capillaries in our hands and feet constrict and blood concentrates in our core, this raises our blood pressure. The increase in blood volume is handle by the kidneys, which kick into action, filtering out water and ultimately increasing the amount of urine produced. Pretty neat.
There are a few other, more mundane reasons we pee more. We’re less active, and when we are active it tends to be colder out, both of which mean that we sweat less during the day. Since we sweat less in cold weather, our body needs to find other means of offloading all that extra water. And since we’re not out hiking, biking, and spending time outdoors, we have easier and more constant access to water. Additionally, the cold dry weather also dries out our skin, which triggers a thirst response. Drink more, pee more.
Also of note is that we excrete more calcium in the winter than in the summer (source). Combine this with lower rates of exercise, weight gain, and a more salty/processed food diet, and we find ourselves at a higher risk of getting kidney stones in the winter.