The reason they have the strongest effect there, and not locally where they're produced or globally in the stratosphere, is because the stratosphere is very dry and only in the extreme cold of a polar winter can stratospheric clouds form. The particles in these clouds catalyse the reactions that allow a each free chlorine molecule to destroy many ozone molecules.
That's also why the hole is worse in the Antarctic. It is colder and has more polar stratospheric clouds.