Anything you don't understand is a good topic for questions.
Decompression theory is, first and foremost :idea: theory. One of the regulars around here is one of the docs who was involved in some of the tests, so he can go into far more detail than I can.
The work of J.S. Haldane was the basis for a model of decompression. His basic idea is that, when you breathe air at higher pressure than atmospheric pressure, some of the nitrogen gas dissolves into your tissues.
When you then decrease the pressure of the gas you are breathing(like when you ascend in scuba), the nitrogen which has dissolved in your tissues starts to be dissolve back into the blood , and is then carried to the lungs where it is exhaled.
How fast the gas dissolves out of the tissue is determined largely by how much the pressure has decreased. The less the pressure decreases, the slower the built-up nitrogen is released. The more the pressure is decreased, the faster the nitrogen is released.
According to this Haldaen model of decompression theory, DCS ("the bends") results when pressure is released "too fast" and your body can't eliminate the nitrogen fast enough. The theory is that if your body can't eliminate the nitrogen fast enough, the nitrogen (and assorted other dissolved gases) start to form little bubbles, which continue to grow in size. In theory, if these bubbles get too big they somehow interfere with nerve signals producing the pain, and other neurological symptoms of DCS. (A certain amount of small bubbles is usually tolerated without incident. These are referred to as silent bubbles.
Making a safety stop at 15ft means that your breathing gas will be a little less than 1.5atm pressure. You are shallow enough than you are eliminating nitrogen at a reasonable rate, but (in theory) not so shallow that the nitrogen will be released "too fast."
Finally, remember that safety stops are done because no dive table is absolute. Occasionally someone who has stayed within the limits of the dive tables and followed all the rules gets a DCS hit. These are what people refer to as undeserved hits. Safety stops should reduce the chance of one of these undeserved hits.