So, step one, find a Taxi cab. You then have a choice to press the up button on the D-pad and start your fine career as a taxi driver. The best time to do this is prior to opening the second island. After certain story events, driving a taxi on the second island becomes pretty dangerous.
Now, the trophy is for dropping off 25 passengers, but being a Taxi driver is tied to the 100% Completion trophy. You must actually drop off 100 passengers for the Taxi job to be considered completed. Luckily, these do not have to be done all at once, or in the same Taxi cab.
If you start this on the first island, you can get 100 passengers dropped off in roughly 1 to 2 hours, depending on their destinations of course. Also, this is a great time to cycle through the radio for the "Born in the 80's" trophy tied to listening to all the radio stations.