If we think that consciousness is simply a process that happens in the brain, and that what makes consciousness going on in my head “me” is certain properties of that process (certain memories, certain ways of thinking, ect) then we must accept that it is possible for consciousness to “branch”. Branching is when there are two (or more) different conscious processes that can legitimately though of as continuations of what used to be a single conscious process. For example, if a complete and perfect duplicate of a person were to be created (down to the last electron) then the consciousness in both the original and the duplicate could be said to be the same consciousness (in the sense of continuation) as the consciousness that was in the original before the duplicate was made. But, since we can’t currently make duplicates of people, we tend not to think about such possibilities. After all I am unlikely to be duplicated in this way within my lifetime. But, if we accept the multiple worlds interpretation of quantum mechanics, duplicates are being created constantly, although not in such a way that they can interact with each other. And this causes some to ask: “why am I this particular person in this moment instead of one of the many other possibilities?”
This question cannot be answered, not because it reveals some deep mystery, but because it is based on faulty assumptions. Specifically it assumes that there is some “I” that inhabits different consciousnesses at different times, basically a form of dualism. Of course such thinking (or at least such confusion) persists because of the way we talk about consciousness; we speak of past and future moments of consciousness as being me, and so come to envision a single self as tying together these moments. But really there is no such thing. Certainly there are connections between these past conscious states, my current conscious state, and future conscious states, but these connections are not the result of sharing some common mental substance.
We can ask: “why am I in the conscious state I am in now?”, but the answer is rather boring, involving details about the current physical state of the brain and how they result in the experience that we are having (certainly the explanation of how they are connected would be interesting, but even I think all the details would be tedious). So in the case of multiple worlds it is not the case that “I” am in one of them and that someone else is in the rest. Let consider a particular moment in a particular world. That moment in that world is followed by a large number of worlds, and in each world (or at least all of the future worlds in which I am alive) there is an “I”, each of whom can equally claim to be the person who existed in the initial moment.
It is because we understand so little about consciousness we are all too willing to give it mystical properties, assuming that somehow the consciousness that is “I” moves between various times and places. And thus, when confronted with the idea that many worlds, all of which contain a consciousness that could be seen as mine, branch off from a single source we sometimes mistakenly assume that my consciousness must pick one branch to go down, or worse, that it goes down all the branches but somehow is the same consciousness in each world, leading people to mistakenly assume that we can know this model to be false a priori because we don’t experience being in multiple worlds at once. In any case the question itself is a good argument against dualism, because under a dualistic world-view we must consider it a legitimate question, which has no answer, which would be a failure of dualism, on its own terms, to explain the world.