Two things, mainly. The common cold is caused by a ton of different viruses. That's why people who have "a cold" could have anything from a mild head cold to an annoying but dry cough to a deep, wet chest cough. It just depends on the specific virus.
More relevant to CF, the other factor is we're already (well, most of us anyway) chronically sick. Our immune systems are already fighting whatever bacteria are colonizing our lungs. Adding another virulent microbe to the mix can overtax the immune system and let the bacteria in our lungs flare up while our immune system attacks the virus instead of fully focusing on the bacteria, which lets the bacteria grow and causes an exacerbation.
Our immune systems are great for singling out foreign microbes to attack, but are lousy with prioritizing which ones are the bigger threat. Pseudomonas or Staph or whatever is way, way more virulent and destructive than a rhinovirus but our immune systems don't know that.