CFCs are relatively inert- they do not participate in chemical reactions. This is a major reason why they are so commonly used. Unfortunately it also means that these compounds persist in the environment for long periods of time. Gaseous CFCs eventually rise into the upper atmosphere where they encounter a very different environment than that near the surface.
Here high energy radiation from the sun impacts the atmosphere and causes a variety of chemical reactions to occur. From the point of view of living creatures, one of the most important is the absorption of ultraviolet (UV) light by ozone, and its CFC molecule, a highly reactive halogen atom, such as chlorine, gets released. This chlorine atom acts as a catalyst, causing the degradation of ozone without the absorption of a high energy light wave. Thus, the effective concentration of ozone decreases and more UV light can impact the surface of the planet.
Because the released halogen is a catalyst, a single CFC molecule can cause the destruction of many ozone molecules. While the ozone is constantly being replenished by other natural processes, the rate of depletion in the presence of large amount of CFCs is faster. this is why many countries have limited the production and release of these compounds, so as to prevent a potentially catastrophic change in the environment.