What "you" are changes with time (consider that you're quite different now from when you were, say, 5). The implication of this is that once a copy is made, new experiences are formed by both copies and their patterns change in divergent ways. If you destroy one after a copy is made, the changes undergone by the destroyed one after the copying aren't transferred to or recreated again, and so are lost. Or in other words, if you make a perfect copy, they're identical at the moment of creation, but virtually immediately afterwards won't be. If you destroy the original before making the copy, then the copy is identical to the original at the moment it is destroyed, ideally, and so the same state last experienced is re-achieved and can develop further.

I'm struggling to think of the proper words to explain my thoughts on this subject, so I'm sure my responses about it are somewhat confusing, and my attempts to elaborate make them fairly long, I'm sorry about that.

Something that I think might be a source of some of that confusion is that I get the impression that many think of consciousness as a distinct nonphysical "thing" that is somehow tethered to the brain, such that the destruction of the brain results in the severing of that connection in a way that means it can't be caught and pulled back again by any physical process, similar to how people that believe in souls posit them to behave.

I do not believe consciousness works this way. I think that it literally is a specific form of information, or perhaps an emergent effect of certain kinds of information processing, and thus, is a part of the physical universe in the same sort of way that a digital image is (the image itself isn't "made" of any substance and can be encoded into any form of matter that can be organized into a sufficiently complex arrangement, but that organization physically exists, changing it changes the image, or produces a new but similar one depending on how you define it, and it cannot exist if no matter exists in an arrangement that can encode it), and as a result of that, getting it back just requires getting some matter into an arrangement that encodes it again. The tricky bit is that unlike a digital image, it isn't a static sort of information but a changing one. So, to take the analogy further, replace the image with a computer program that takes inputs from the world around it, and then rewrites it's own code in response to those inputs. If you take this algorithm, pause it, copy it's state and destroy the original machine while rewriting that state into a new machine in a new location, and unpause the program on the new machine, you'd get the same results as if you had just paused it, moved the original machine to the new location and unpaused it at the same time you would have unpaused the copied program. There's no basis to say that you have a different program, because they have the same code and are behaving the same. But if you unpaused the original machine, its instance of the program will change itself, and then if you destroy it now, the copy won't reach the state that that last version of the original would have reached had you brought it to the new location too. In this analogy, killing a person is equivalent to one of these programs reaching a state that is no longer continued, so if you continue it later, somewhere else, even on new hardware, that's fine, and if you create a branch and keep both running, that's also fine, but if you create a branch, and then destroy one without recording it's state to recreate it later, or just never actually run it again on a new machine, that branch has reached an end state that doesn't continue changing itself, and so you've had "someone" die.