Two possibilities.
1) There exists a soul.
In this case a soul is magic enough that it comes with you when you transport.
2) There does not exist a soul.
In this case, consciousness is an illusion, all that matters is the perception of consciousness, and the fact that your perception is continuous means that you are continuous.
What if perception isn't continuous, though? What if it's like a film, where individual frames move fast enough to simulate the appearance of seamlessness? We live in a universe where the perception of time moves at a variable rate dependent upon many factors.
The perception of any observer is flawed. Knock a man unconscious without his knowing it, and when he recovers, he will not have noted any time passing until he has a chance to gain awareness of his surroundings. For him, no time has passed until that very moment, but for everyone else around him, the perception of time moving forward is noted and logged. The observer, too, has experienced the forward motion of time, as his body's cells will attest, even though he was not aware of it.
Perception itself can be an illusion.
So a person can be replicated, and the original killed at the moment of transmission, all without being noted by the observer, because the observer has no knowledge of what just happened other than the perception that a smooth transport has taken place.
Of course, that's assuming the machine could instantaneously, and I mean literally instantaneously, assess, break down, transmit, and reconstitute every quark and lepton in a person's body elsewhere, along with the immediate atmosphere surrounding it, without killing that person and everything around that person.