There is no way to be "sent" anywhere because there is no soul to be shuttled around in the first place. There is only a sequence of experiences that are correlated with each other. In the dead branch there will be no experience, but in the living branch there will be, so the "illusion" of continuity (which is really just correlation) will be allowed to continue there.

Full disclosure: I haven't the slightest clue how quantum mechanics works.

The idea is that as long as the state-machine-which-is-you continues from its last state somewhere, your consciousness survives (since that state machine running is you being conscious). Quantum suicide splits the state-machine-which-is-you into n state machines, and at the same time destroys all except 1.

That means there will be a state machine continuing the same computation from the last state, which is just another way of saying that your consciousness survives.

It all depends on the ontological status of other Everett branches. If they're truly real (and not just something that appears in the calculations), quantum suicide should work.

The way I think about "self" is that there is no immaterial soul or anything like that. There is just the "experience of being you". Everything that has this experience, is you. And when there is nothing to have this experience, you are dead.

The "experience of being you" is also not defined exactly. Is it still you, if we replace a position of one atom in your body? Sure; that happens all the time. Two atoms? Yep, the same thing. Shuffle all atoms in your body? Nope, that would almost certainly kill you. Move all atoms in your body in a way that creates an atomic-level perfect replica of Elvis? Nah, we just killed you and re-created Elvis. Okay, so how many atoms can we move? Ehhh... no precise answer, just a "fewer is better than more" heuristic.

For practical purposes, the new experience of you is the same "you" as the old experience, if it shares the memory (again, the memory is never perfect, but "more is better than less") and has the same character traits (again, "more is better than less"). Under usual circumstances, this usually happens. The weird cases are people with serious brain damage -- the others may describe them as "no longer the same person".

So, according to the many-worlds interpretation, the universe is branching all the time, and all branches contain the real "you" -- a body that "experiences being you" and shares the memories and traits with the past you. You subjectively never feel these multiplications, because each copy is in a paralel reality, unconnected to the other copies; from each copy's perspective, it is the only one that exists.

(Assuming a sci-fi technology, some kind of atomic replicator, it might be possible to create multiple copies of you in the same reality. Even then, each copy would be the real "you", unconnected to the other copies -- that is, maybe seeing them from outside, like other people, but each copy from this point onwards would experience its own story.)

The idea of quantum immortality is that if you die in one branch and not the other, then the lucky branch is the only one containing a body that "experiences being you". That copy feels perfectly alive, and it is the real "you". -- And the other reality contains a dead body that experiences nothing.

Considering that the universe was already branching millions of years before you were born, the vast majority of realities does not contain "experiences of you" anyway. Apparently that does not bother you in everyday life... so, analogically, the surviving copy of "you" should not be bothered by the fact that some other branches no longer contain you. Also, the universe was already making your copies since you were born... and some of those copies have already died... and yet it does not make you feel less alive.

So, the idea of quantum immortality is that that in the future, some of your copies will survive, and they will not be subjectively bothered by the diminishing number of realities they exist in, because subjectively, your reality is all there is.

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What do I think about it? I think this is probably correct from some perspective, but less encouraging than it seems, because I believe that the exact fraction of realities that contain "you" is important in some sense. Despite already being "one in a zillion" (the probability that Earth contains life, dinosaurs are extinct, and the correct human sperm impregnated the correct human egg -- all apriori extremely unlikely things), there is a significant difference between "one in a zillion" and "one in hundred zillions", and in the latter case, you are still just as much alive from the inside view, but less alive from the outside view, and -- for reasons I am not sure I could articulate properly, and I am not completely sure about their consequences -- the outside view should still matter to you.

So, while quantum immortality can be a happy thought (or maybe an unhappy thought; no one says that those immortal lives are necessarily nice), I will still take care to protect my life in the ordinary sense, to keep myself alive in as many realities as possible.

"subjective" is itself subjective.  There are entities experiencing all the things. If you believe that other beings can have qualia, these virtual-copies-of-you have qualia.  Whether they're the "same" entity is dependent on your ideas about identity.

I don't know anyone who claims that it'll be a linear or unified experience.  Without continuity and communication across instances, I don't think of it as personal immortality in the simple sense, any more than I think about children or great works as immortality.  

Woody Allen had it right: 

“I don't want to achieve immortality through my work; I want to achieve immortality through not dying. I don't want to live on in the hearts of my countrymen; I want to live on in my apartment.”