Transporters work by de-assembling something (e.g. you) and re-assembling it somewhere else. What if, when you’re dis-assembled, you die, and the re-assembled version of you is essentially a copy? Then every time someone steps onto a transporter, their final thought before death is that they’ll end up beamed somewhere else. And the re-assembled version (copy) just thinks that everything went fine and continues on like nothing bad happened.
It is still you, just not the same you that you were before being killed and cloned. Look at it this way, if the technology existed for me to make a perfect clone copy of you with all your abilities, memories, and flaws exactly as they are, but in order to do so I would need to remove and liquefy your brain but otherwise leaving your body intact, then the you that you currently are would cease to exist. To the outside world and everyone in it, the clone of you would still be considered you as it would be indistinguishable from the original, but the you that was you before the cloning process is now a brainless corpse back in my lab.
The original version of me ceases to exist, but for all intents and purposes there has no been no break in existence for [OmegaMouse]. You could say that those are two separate people, but what if the same happens every time we go to sleep, and we’re none the wiser? As long as someone identical comes out on the other end, what difference does it make?
It makes no difference in the sense that a version of you would still exist, but for the version of you that ceases to exist, I’d think it would matter to them. The movie The Prestige from 2006 somewhat illustrates my point on this.