I think I understand the arguments, but I've never really understood at a fundamental level why this concept is so disturbing to many people. If you destroyed me and instantaneously made an exact copy, then I continue to survive in every meaningful way. You have to dip into some seriously non-materialist, entirely unsupported spiritual beliefs to claim otherwise.
I have no reason to believe my consciousness or the world itself is continuous, except that it seems less complicated than the alternatives that I can think up. What's more, given the fraction of my life during which I have no remembered experiences, it seems like I experience deaths every day which are a lot more serious and long-lasting than a momentary death on a transporter pad. If stepping onto a transporter means destroying every cell in my body and creating an exact copy. . . that's fine. Why wouldn't it be?
If you do believe in an immaterial soul. . . then, how does a transporter lock on to your soul and insure that it remains with your body?