Heisenberg compensators remove uncertainty from the subatomic measurements, making transporter travel feasible. Further technology involved in transportation include a computer pattern buffer to enable a degree of leeway in the process. When asked "How does the Heisenberg compensator work?" by Time magazine, Star Trek technical adviser Michael Okuda responded: "It works very well, thank you."
Anyway lets for the sake of argument assume that the transporter does work. What Skinny Pete was saying to Badger in Breaking Bad was not an argument about the physics but an argument from the philosophy of mind. Pete's contention was that every time Kirk gets transported somewhere it's not the original Captain Kirk but a recreation, a copy, a copy who has the same mind state and the same memories as the about-to-be transported individual but it's not the same man. The original Captain Kirk, Pete argues, is destroyed in the transporter room and a copy reassembled at the transport site. Consciousness, Pete is implying, cannot be transported because consciousness is a property that cannot be subject to measurement. The copy of Captain Kirk now on the alien planet only thinks he's the same man because he shares Kirk's memories and mindstate at the moment of transportation but unfortunately the real Captain Kirk was killed by the transporter. The copy goes on to live Kirk's life, Pete says, until he too must enter the transporter and he too dies. Throughout the course of the show, Pete argues, there are hundreds of Kirk copies that get created, while the real Kirk dies the very first time he gets transported. Pete just throws this idea out there and then Badger pitches his Star Trek pie eating contest script idea but its worth thinking about. If you want to know more about the concept there's a Star Trek wiki discussion on what happens when you transport, here. And there's a really nice philosophical analysis of the transporter concept by Douglas Hofstadter and Daniel Dennett, in their 1982 book The Mind’s I where they call the transporter a "murdering twinmaker." Hofstadter further explores consciousness in his fascinating book I Am A Strange Loop and Daniel Dennett examines in some depth the idea of materialism in his - rather misleadingly - titled book Consciousness Explained. Whether you believe in strict materialism or a Cartesian dualism I'm not sure I can see how the transporter could work without killing you and replacing you with a copy.
But what exactly is consciousness and how does it evolve from matter? It's a huge problem in philosophy, psychology and neurology. Antonio Damasio attempts to explain some of the elements of the problem here: