I'd like to hear the argument against this view, rather than dismissals of it as fantasy and nonsense. It really isn't. There's a compelling case for it being true, given a couple of relatively reasonable assumptions.
But first you need to understand it.
It depends on two key ideas.
The first idea is that existence is not finite. We don't need to appeal to a quantum multiverse or any of the more exotic parallel universes. All we need is for space to go one forever -- for there to be no border to space, and for space not to repeat itself periodically or to wrap around as does the surface of a sphere.
As far as we can measure, the curvature of spacetime is flat. If it is precisely flat, then it would seem this assumption is true. Whether or not it is actually true is debatable, but the idea is neither fantasy nor nonsense. However if it is true, then there are certainly an infinite number of people indistinguishable from you to be found if you just travel far enough. They live on a planet just like ours, and have had the same life experiences and recorded the same memories.
Greta has offered some more detailed thoughts than "fantasy and nonsense", thankfully.
While a theoretical infinite reality can theoretically produce infinite things, that means there's an infinite number of possibilities that have not yet occurred.
I don't think that follows. A finite reality would mean that there are an infinite number of possibilities that have not yet occurred. It's harder to see this with an infinite reality. If space is infinite, then anything that can have occurred since the Big Ban will have occurred somewhere. However, there are an infinite number of things which cannot yet have occurred, I guess, simply because there hasn't been time yet. Although this kind of assumes that there is such a thing as a global universal time across all of space, the idea that it makes sense to talk of what's happening right now on the other side of the observable universe, and in general relativity that would appear not to be the case (though there are physicists such as Lee Smolin who would beg to differ).
In fact, statistically there should be a tiny number of people who are extraordinarily similar to you, then there'd be more who are not quite so much like you, more again who are even less like you, and so forth.
Not quite. If space is infinite in extent, then the cardinality of all such classes of people should be infinite. Infinite people extraordinarily similar, infinite people who are not quite so much like you, infinite people who are even less like you and so forth. Instead of the numbers of such people increasing as you relax the similarity criteria, you should speak of the density of distribution of such people increasing. If you relax it enough, you are likely to find someone sufficiently like you even on the same planet. Relax it further, you may find many such people living in your town or city. But if you keep it very strict, you can expect to have to travel for many billions upon billions of light years (actually I suspect that's rather underselling the distances involved) to meet your nearest doppelganger.
That would seem less like an afterlife than a reminder that we are perhaps not quite as unique as we imagine.
It isn't really an afterlife in any sense, in that the idea of an afterlife is that you go somewhere else (very different, usually) when you die. So I don't agree with that description of the idea (not your fault, but the article's). The idea is instead that we are all immortal, that we can never die. We don't go anywhere when we die, we just keep on living, at least from our own subjective point of view. We still see those around us dying. This makes the idea rather less comforting than actual afterlife beliefs. Indeed, it is chilling.
But to explain this, you need to understand the second big idea, which is a certain account of personal identity. I'm not going to try to defend this view too much, just describe it.
The idea here is that we ought to think of personal identity as attaching to a mind, i.e. a particular life story, personality, set of memories, etc, and not to a physical object (the body). So, if I lose an arm, and get a prosthetic arm to replace it, I'm still me. All the atoms in my body are replaced with other at regular intervals, but I'm still me. If my brain were to be transplanted into an entirely new body, I would think of my personal identity as going along with it. And if parts of my brain were replaced with electronic prostheses that performed exactly the same functions, I think I would still be me -- even if the whole brain were replaced.
Or think of the Kirk transporter problem. The transporter effectively destroys the original body and builds a replica at a remote location. On traditional notions of identity, the original Kirk dies and a new Kirk, an impostor, is born, and so we should not use the transporter if we don't want to die. But on the mind theory of personal identity, Kirk simply moves to a new location. This is because the mind of the teleported Kirk is identical with the mind of the original Kirk. The same life story, personality, set of memories etc continues to exist, but now it is in a new location.
The problem with this way of understanding it is when we are invited to consider what happens when the transporter fails to destroy the original Kirk. Both Kirks can't be Kirk, right? Well, I say they can. Think of a pair of identical twins. Which one of these twins is the original zygote and which is the clone? That question is nonsense. Both are equally descended from the original zygote. Neither is any more an original or a clone than the other. The identity of the original zygote has simply split. These new individuals are not identical to each other, but they are each the inheritors of the identity of the original zygote. The same is true of Kirk, in my view. Each Kirk has equal claim to Kirk's identity, while they are from this point on distinct people.
It's not unlike encountering a fork in a river. Each fork is still the river, but each fork has its own identity from that point on.
From the point of view of the pre-teleportation Kirk, he has two futures, and should one of those futures terminate prematurely (as in when the teleporter functions correctly), he will still be alive in the other future and so he should not be too concerned (unless one of those deaths is preceded by suffering of some kind, including fear and anticipation of death).
Anyway, from this point of view, you are just the pattern that your mind makes, and you exist wherever that pattern is instantiated. If there are infinite copies of this pattern scattered throughout existence, then you are not any particular one of them. You are all of them equally. You perhaps perceive yourself to be sitting at a desk, reading this post, with light coming in from a window. It feels like this experience places you at a very specific time and location, but if this precise experience is not unique it really doesn't. You could be in any one of an infinite number of similar places and times throughout the universe. So I say you are not in any specific place. You are in all of the places (and at all of the times) that conform precisely to your experience. Should any one of the physical instances of you die, then you continue to exist in all of the others.
Due to the anthropic principle, you will of course always find yourself living on a planet where you have not died yet. You can never witness yourself dying after all. So you will appear to yourself to be immortal. Your family and friends will eventually see you die on 99.9999999999...% of planets on which you are instantiated, but there will always be a diminishing fraction (though still infinite number) where your death has not happened yet for whatever reason. As long as it is at all physically possible for you to be alive, then you will be alive somwhere, making you effectively immortal.