CFC's (Chloroflourocarbons) are pretty inert at ground level, but can harm the Ozone layer in the Stratosphere by catalysing it's destruction.
In the first stage, a CFC reacts with UV radiation to form a Chlorine Radical (which has an unpaired electron).
CCl2F2 + UV --> CClF2 + Cl*
In the second stage, that Chlorine Radical reacts with Ozone to produce Oxygen Gas and Chlorine Oxide.
Cl* + O3 --> O2 + *ClO
This is the actual destruction of Ozone. However the real problems are caused by the next stage.
In stage 3, ClO reacts with an Oxygen radical to produce more Oxygen gas and the initial Chlorine Radical.
*ClO + *O* --> O2 + Cl*
It reacts with Ozone (it just depends on which it happens to collide with in the atmosphere). The Chlorine Radical is still reformed this way.
*ClO +O3 --> 2O2 + Cl*
As you can see, the Clorine Radical doing all the damage is regenerated. This means that one CFC molecules can destroy lots of Ozone and this is why the issue is so vast.