I happened upon the website of a Norwegian physicist named Kim Øyhus who independently came to the many-worlds conclusion in 1990. The page strikes me as an unusually good example of epistemic rationality. He starts from a premise ("if the math of quantum mechanics is true,") and moves onto four hypotheses (which cover all possible states of reality, since the fourth hypothesis is, "something else"), figures out testable predictions of each hypothesis, and comes to the conclusion that the Many-Worlds Interpretation is correct.
This is one of several shortened indices into the Quantum Physics Sequence.
Macroscopic quantum superpositions, a.k.a. the "many-worlds interpretation" or MWI, was proposed in 1957 and brought to the general attention of the scientific community in 1970. Ever since, MWI has steadily gained in popularity. As of 2008, MWI may or may not be endorsed by a majority of theoretical physicists (attempted opinion polls conflict on this point). Of course, Science is not supposed to be an opinion poll, but anyone who tells you that MWI is "science fiction" is simply ignorant.
When a theory is slowly persuading scientists despite all academic inertia, and more and more graduate students grow up familiar with it, at what point should one go ahead and declare a temporary winner pending new evidence?
Reading through the referenced posts will give you a very basic introduction to quantum mechanics - algebra is involved, but no calculus - by which you may nonetheless gain an understanding sufficient to see, and not just be told, that the modern case for many-worlds has become overwhelming. Not just plausible, not just strong, but overwhelming. Single-world versions of quantum mechanics just don't work, and all the legendary confusingness and mysteriousness of quantum mechanics stems from this essential fact. But enough telling - let me show you.