People probably started getting it when the weather got cold, and then assosiated it with cold weather.
Why do you suppose that the common cold ( the virus ) came to be called a cold?
The on-line etymological dictionary gives this:
COLD: "coldness," from cold (adj). Sense in common cold is 1530s, from symptoms resembling those of exposure to cold; compare earlier senses "indisposition caused by exposure to cold" (early 14c); "discomfort caused by cold" (c. 1300).