How do CFC's destroy ozone?

That's the question this somewhat complicated figure
tries to explain.  CFC's mainly enter the stratosphere
in the tropics, where convective activity and rising
motion inject tropospheric air in the stratosphere.
Once in the stratosphere, they are exposed to more
powerful ultra-violet (UV) radiation, which is screened
out in the troposphere by all the ozone molecules above.
Freed chlorine atoms react quickly with O3 (ozone) or
CH4 (methane) to form ClO and HCl respectively.  In the
former case, the ClO can react either with itself to form
ClOOCl (the dimer) or NO2 to form ClONO2 (chlorine nitrate).
ClONO2 and HCl do not react with O3, and are therefore
said to safely sequester chlorine as reservoir species.
ClOOCl is an unstable molecule, capable of being broken
apart by sunlight, and so is not found in large quantities
during the daytime.  Therefore, the catalytic cycle
Cl + O3  -> ClO +ClO -> ClOOCl + hv (sunlight) -> 2Cl
does not occur in the tropics or midlatitudes to the same
degree found in the Antarctic polar ozone hole, since most
of the chlorine in these regions of the atmosphere ends
up in one of the reservoir species (HCl or ClONO2).

In the Antarctic, however, a combination of unusual circumstances
tilts the balance of chlorine away from the reservoir
species and towards ClO and ClOOCl.  First, the Antarctic
polar region is highly isolated dynamically during the
Southern hemisphere winter season, moreso than the Northern
polar air mass during its winter.  The high degree of isolation
during the dark winter months allows the temperatures
over Antarctica in the lower stratosphere to drop well below
200 K.  When temperatures fall below 195 K, nitric acid trihydrate
clouds begin to condense from H2O and HNO3 (nitric acid).  When the
temperatures fall below 188 K, water clouds condense as well.
These clouds provide surfaces on which an important reaction, which
doesn't occur in the gas phase, occurs.  On the surface of these
clouds, HCl reacts with ClONO2 to form HNO3 + Cl2.  The Cl2
then breaks down in sunlight to free the Cl atoms, which then
take part in the catalytic cycle to destroy ozone.

Because PSC's tie-up the nitrogen species, the frequency of the
reaction ClO + NO2 is diminished due to the lack of NO2 molecules.
When the temperatures begin to rise as the sun comes up over
Antarctica in the spring, the clouds evaporate, freeing up the
nitrogen.  This permits ClO to react with NO2 again, forming
ClONO2.  Without the surfaces present, the chlorine gets trapped
in the reservoir species again and the catalytic cycle shuts down.

If the temperatures are cold enough for long enough during the winter,
crystals in the PSC's can grow large enough for nitrogen to precipitate
out of the stratosphere, thereby reducing the the flow of chlorine into
ClONO2 even after the temperatures have increased and the clouds have
gone away!

This chemistry will only occur in regions where temperatures are cold
enough for long enough periods of time that PSC's form and persist.
This helps explain why the ozone hole forms in the Antarctic polar region
during late winter/early spring.  And why the ozone hole only exists
for a short period of time (6-10 weeks) before recovering.