First off, I don't mean recreational peeing. We're talking about honest-to-goodness needing to piss or else, and if those of us equipped for it can draw in the snow, hey, that's neat too. The phenomenon of needing to pee in the cold has a name: cold diuresis. It remains to be definitively accounted for by doctors, but several explanations seem fairly reasonable.
When you get cold, your blood vessels, particularly those in your fingers and toes, constrict because of something called vasoconstriction. In an effort to maintain a warm core temperature, the body tries to keep blood away from more susceptible extremities. Simply: the body lets less warmth out, and less cold in, and one way to do that is by not allowing blood into colder places. This is accomplished by limiting the space that blood could occupy. I've know people with Raynaud's disease, a disorder in which the body exaggerates this effect. Their hands would turn all sorts of blue and pink and be really cold (to them and to the touch). The effect is startling.
How this theoretically works for your bladder is this. Because you're sending less blood out to the extremities by reducing the volume it can occupy in those extremities, you have more blood elsewhere. The same total amount, but less space—this, naturally, equals higher blood pressure. To regulate that, your kidneys move to pull liquid out of the body, which leads to more liquid in your bladder than there would normally be. Sup, pee.
That's the dominant theory, but there are others. One is a bit more technical and has to do with proteins called aquaporins. As the name very smoothly alludes to, these proteins exist to form channels between the inside of your cells and the outside for water to travel more rapidly. In cold weather and/or under the influence of booze, these are inhibited in certain places around the body, like the brain and kidneys. When water isn't pouring into those cells, it remains in the bloodstream, where all of the sudden there is too much of it. Hello, pee.
Or maybe you're drinking to stay warm like a damn fool. Heyooo, pee. Don't do that. Or rather, do that, just not if you're in actual danger of hypothermia because it will just make the situation worse. Hello, death.