There's a couple of reasons for it: First is, your nose has a ton of tiny blood vessels that it uses to support all the glands you use to smell. When it's cold out, these blood vessels expand, sending more blood to your nose. This makes your nose produce more fluids.
The second reason is that, since the air you are breathing in is colder, your nose needs to work overtime in order to warm it up. For your nose, working overtime means making more snot and making it drip out of your nose.