Answering the “how we knew” part: it was an indirect (yet convincing) link between CFCs and the observed decrease in ozone concentration (which had been discovered by spectroscopic measurements).
Investigating the ozone hole by sampling the atmosphere with high altitude research aircraft, it was found that O3 concentration was negatively correlated with chlorine monoxide (ClO) concentration. The only plausible source for increased amount of chlorine there is the CFCs (which release Cl atoms upon hit by high energy solar radiation, present at high altitude). And there was a known chemical pathway for ClO to destroy O3. Therefore, the conclusion was that the CFCs were the cause for the ozone depletion.
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