Normally the air in your lungs is warm but when you breath in that cold air the cold air rushing into your warm lungs causes a lung spasm which is normal which then causes your lungs to hurt and makes it hard to breath.
I notice that my lungs hurt and it's very hard to breath when I'm outside working in below zero temperatures.
If you have asthma the cold air can trigger an asthma attack so it's best to avoid being out in the really cold temperatures if you have asthma.