Quantum Mechanics is a curious area of study which began in the early 20th century when scientists began to discover that the theories of electromagnetism and Newtonian mechanics, which so elegantly describe the movements of normal objects, completely fell apart at extremely tiny atomic and subatomic scales. It soon became clear that a separate theory would be necessary to describe subatomic interactions, and thus Quantum Mechanics was born.
The theory of quantum mechanics describes a tiny realm completely foreign to the one we observe normally. At quantum levels, matter exists simultaneously as particles and as waves (wave-particle duality), a particle's position and momentum cannot be precisely known at the same time (Heisenberg uncertainty principle), and the state of two objects can be intertwined, regardless of the physical distance between them (quantum entanglement). Niels Bohr, one of the fathers of quantum mechanics, once said, "Anyone who is not shocked by quantum theory has not understood it."
The predictions of quantum mechanics have never been disproved in any experiments in over a century of development. It has been studied by brilliant minds including Albert Einstein and Richard Feynman, and though there is much disagreement about what it all means, there is little doubt that it is true. Some even think it provides us with a means to live forever.
Quantum mechanics is not in the business of exact predictions, rather it deals in probabilities when describing the position or momentum of a given particle at a certain time. This inexactness is not because the theory is incomplete, but because those qualities of a particle are inherently unpredictable with any precision; or to put it another way, because there seems to be some degree of randomness at play in the universe. Einstein was famously uncomfortable with this facet of quantum physics, asserting that "God does not play dice!" But despite spending a good deal of his life after 1925 trying to back up his assertion, he was never able to.
In 1957, a student named Hugh Everett suggested that perhaps the reason that a particle's outcome can't be predicted is not because of randomness, but because every possible outcome does occur. This idea led to the "many-worlds interpretation" (MWI) which postulates that at the quantum level, everything that can happen does happen, and that each possible outcome branches the universe into another which is at first identical aside from the alternate outcome. So the seemingly "random" outcome is actually just representative of the one possible outcome one's current universe happens to be based upon. The overlapping universes, between which no information can pass, would then continue to develop individually, each of them branching endlessly as well. Among physicists worldwide, this "multiverse" idea has become one of the most widely accepted interpretations of quantum physics.
On a larger scale, MWI would mean that everything which can happen will happen in at least one universe. Based on this, Max Tegmark at Princeton University suggested an experiment to prove that the many-worlds interpretation is correct, where one points a loaded gun at one's head, and pulls the trigger. If you were to try this test, it is highly unlikely that you would survive... but if the gun failed to go off, and continued to do so in subsequent tests, you could eventually become reasonably confident that you're in one of the branched universes where something caused the gun to misfire each time. Of course only the "you" in those "miraculous survival" universes would know this, the others would all be dead from gunshot wounds to the head.
Obviously this ridiculous test is not recommended, but if MWI is true, wouldn't that mean that there are universes where I decided to try this for myself, and in at least one of those universes I survived miraculously? This article is probably much more exciting in those universes. Incidentally, in this branch, my brain is currently trying to gnaw its way out of my skull in self-defense.
In such a way is the argument for Quantum Immortality made. Some say that regardless of the cause of death, if the many-worlds interpretation is true, then there will always be at least one branch where the "miraculous survival" scenario is realized, and that version of "you" will never die. Of course the odds are overwhelmingly against the possibility that anyone in this universe is a perpetual miraculous survivor. Although the whole idea is wildly speculative, quantum immortality violates no known laws of physics.