What does the ozone layer do?

Short Answer: The ozone layer protects the DNA of all surface dwelling life, by absorbing UltraViolet-B from our Sun.

More detail
The ozone layer is the product of very short wave Ultraviolet light (UV-B) breaking apart oxygen molecules. When ozone is formed, it serves to block energetic UV (UV-B) that other molecules don't, and acts to moderate day / night temperature swings a little bit.
Ozone absorbs UV-B, which would otherwise damage our DNA, and the DNA of all surface dwellers on Earth (including phytoplankton in the oceans). This absorption of UV-B or more energetic light breaks the ozone down into an oxygen molecule and a single oxygen atom. A few of these fragments reassemble as ozone. But primarily UV-B is required to break apart more oxygen to make replacement ozone.

The ozone layer is like a blanket that shields Earth (not just the living things). It is a "greenhouse gas" that moderates day / night temperature swings, and allows the "heat" of the nitrogen and oxygen at altitude to be radiated away (both up and down).

Diseases that increase as the ozone layer gets thinner...
  • cataracts
  • skin cancer (melanoma)
  • cancer in general
  • malnutrition / starvation (since our food crops are similarly attacked)