Basically, ozone removes odor by destroying the molecules, bacteria, and spores that cause unpleasant smells. Ozone (O3) is a highly reactive molecule and is very quick to chemically react with particles that it comes into contact with in the air and on surfaces. The extra oxygen atom in the ozone attaches itself to other molecules, chemically changing their structure to create non-offensive molecules – eliminating the smell.
When considering bacteria and spores the same process applies, with ozone attacking the outer shell of the cell causing it to break down. When the outer shell degrades, the cell itself breaks apart and dies.