Why does the cold make us sick?
it does not, really. unless we're talking about things like frostbite or severe hypothermia.
it's a myth that it does.
the virus is more stable in colder air, however.
see more here:
Study Shows Why the Flu Likes Winter
Influenza Virus Transmission Is Dependent on Relative Humidity and Temperature

Innate responses proved to be comparable between animals housed at 5 °C and 20 °C, suggesting that cold temperature (5 °C) does not impair the innate immune response in this system.