Passenger transports allow you to take several people into your ship and ferry them to other locations. If they're satisfied with the speed of the move, they will reward you accordingly.
To obtain passengers, visit a passenger transport facility and type REQUEST. If you have the requisite points (350 at time of writing), you will be prompted to choose between a local passenger dropoff and a remote passenger dropoff. Local passengers will only request transit to planets found in numbered sectors, while remote passenger transports require that you fly your ship to a planet found in uncharted space.
To figure out where each passenger wants to go, talk to them. If, say, you're transporting a dirty man, you would type TALK TO DIRTY MAN. He will then present you with a menu of conversation topics, one of which should be "Where are you going?" If you're transporting passengers out of the numbered sectors, the passenger will attempt to give you directions to the location they wish to go to.
Dropping Passengers Off:
Dropping the passengers off is a simple matter of landing on the planet they wish to go or docking with a space station. They will automatically pay you (assuming you're in the control room) and leave the ship.
Should you decide you no longer wish to transport the passengers, you can type DISMISS <passenger name> to dismiss them from your ship. You must be landed to do this.