How does the ozone layer "stay in place"?
For the ozone layer to dissolve, it would need something to bind to. Chloroflurocarbons are one very notable thing that it can bind to, but that doesn't exactly answer the question.
While the ozone layer would be 3 mm thick if it was compressed down to the pressure of earth's surface, since it is further up in the atmosphere, it can take up a larger volume (like 20 km thick). The ozone layer is maintained by the Chapman cycle.
That is, the oxygen molecule ($\ce{O_2}$) is dissociated by UV radiation (photolysis) to oxygen atoms ($\ce{2O}$). Then the photolyzed oxygen atoms ($\ce{2O}$) can bond with two oxygen molecules ($\ce{2O_2}$) to make an two ozone molecules ($\ce{2O_3}$), provided another molecule can take away some excess energy.
Since it is a cycle, there has to be a way for the ozone to return to oxygen. One ozone molecule ($\ce{O_3}$) can photolyze to make an oxygen atom ($\ce{O}$) and oxygen molecule ($\ce{O_2}$). The oxygen atom ($\ce{O}$) can react with another ozone molecule ($\ce{O_3}$) to make two oxygen molecules ($\ce{2O_2}$).
To just answer your question directly, the ozone layer is replenished by the sun's UV energy reacting with the oxygen in the atmosphere.