The cold air outside is dry air which can irritate your lungs especially if you already have lung conditions or other breathing problems and cause your throat to dry out which makes your throat hurt.
Dry air causes a lot of sore throats and when you're breathing in that cold dry air instead of moist air it leads to a hurting throat.
Besides causing sore throats the dry air can also cause shortness of breath and the cold air can also cause bronchial tightening or constriction leading to coughing and wheezing.