Xaq Rzetelny in Ars Technica:
According to the Star Trek: The Next Generation Technical Manual, when a person steps onto the transporter pad, the computer uses “molecular imaging scanners” to scan his or her body, before the person is converted into a “subatomically debonded matter stream.” In other words, a crew member is taken apart piece by piece, breaking apart the bonds between individual atoms. Then, particles are streamed into a “pattern buffer," where they remain briefly before being sent to their destination.
This sounds an awful lot like death. In fact, it’s even more death-y than conventional death where, after the body’s processes have stopped, the body slowly decomposes. The effect is the same—the pieces of you come apart—the transporter’s just a lot more efficient at it.
Once the matter stream arrives at its destination, the person is somehow “rematerialized” or put back together. While the transporter tends to use the person’s atoms to reconstruct a human, it really doesn’t have to. The machine could use totally different atoms, and the effect would be exactly the same.
In fact, in the Deep Space Nine episode “Our Man Bashir," Captain Sisko and a few other officers are nearly lost during a transporter accident. They beam out from their sabotaged runabout at the last second, but the transporter malfunctions and their patterns must be sent into the station’s computer somehow to save them. Their physical bodies are saved as holographic characters in Dr. Bashir’s holosuite program. Later in the episode, they’re reconstituted using the patterns stored in the holodeck—almost certainly with entirely new atoms.
That sounds an awful lot like a copy—or like a new person. If the transporter is just scanning your data and creating an identical copy somewhere else, then by any reasonable definition, the original person is dead. By analogy, consider a car model. Many cars are produced by the same manufacturer, all from the same design. There’s no way to tell these cars apart, but they’re not the same car.