A TRSA is an area where pilots have the option of receiving Radar service in a location which has not qualified as a Class C or Class B. As mentioned in @ryan1618's answer, a TRSA does not change the classification or regulatory nature of the airspace.
I had the creation and operation of TRSAs described to me at an Oshkosh forum by an airline captain. He explained it something like this:
A TRSA comes about when a radar system is upgraded, and the older unit is appropriated by a congressman for his home district. He gets it installed at his local Class D airport to the delight of the airport and tower managers. > When calling the designated approach control, if a pilot is not interested in having the controller tell him what to do, he need only say "negative radar service." If inbound to land at the delta airport, they may call the tower and hear a voice remarkably like that of "approach control"
If I were a Class D tower operator. I would think it was cool to have radar coverage at my airport, and it might make me feel like a big-shot to act like I was a TRACON facility. Indeed, it may even be a safety enhancement and provide a training opportunity to the tower controller.
In my experience, there are some radar equipped Class D airspaces which operate like this without having a TRSA charted. A clue lies in the existence of an approach frequency in the chart supplement.
- KFSD Sioux Falls, SD
- KFLO Florence, SC