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The main point is that fluorocarbons are catalysts: they greatly speed up the processes by which ozone breaks down in the atmosphere, and because they are not themselves destroyed in ozone-breaking reactions one molecule of CFCs can help destroy a lot of ozone molecules.
In a little more detail, the process of accelerated ozone depletion begins when chloro-fluoro-carbons [CFCs] and other ozone-depleting substances leak out into the atmosphere. Winds quickly distribute the gases evenly throughout the lower atmosphere. CFCs are extremely stable, and they do not dissolve in rain, so they stay in the atmosphere and some of them gradually gain height. After a period of several years, CFC molecules reach the stratosphere, about 10 kilometers above the Earth's surface, where the ozone layer is.
Strong ultraviolet light breaks apart the CFC molecules. CFCs release chlorine atoms, and it is chlorine (and its cousins flourine and bromine) that destroy ozone by pulling the third oxygen atom away from the other two in the ozone molecule. One chlorine atom can destroy up to 100,000 ozone molecules before it finally falls out of the stratosphere
Ozone is constantly being produced and destroyed in a natural cycle in the stratosphere. What the presence of chlorine does is it accelerates the destruction process, and so the concentration of ozone falls and stabilizes at a lower level--the more chlorine, the lower the level.
Since ozone filters out ultraviolet radiation, less ozone means higher ultraviolet at the surface. The more depletion, the more skin cancer, cataracts, damage to materials like plastics, and harm to crops and sea life.
This threat is not something that life has evolved to cope with...