In a nutshell: the multiverse makes it possible for our universe to be described by an beautiful theory.
Why does beauty imply a multiverse? Because of two things. First, beautiful theories seem to be quite rare. The concept of theoretical beauty involves unique and highly constrained combinations of mathematical structures, combinations which are rarely discovered and whose discovery usually heralds revolutions in both mathematics and physics. And second, if we rule out multiverses then each theory can correspond to exactly one universe. Each theory has just "one shot" to get it right, producing the conditions for life. What's the chance that one of the very rare beautiful theories could also jump this very high hurdle? It seems pretty small to me.
But with a multiverse theory there is no problem. We can have a fantastically beautiful theory that has no adjustable parameters at all - for example, string theory. And this theory can also accomodate the existence of life, in some of the many allowed paths in the evolution of the multiverse.
But what about the lack of "explanatory power"? Haven't we given up the most important thing that a "scientific" theory is supposed to have?
Not at all. Take a look at any of the various definitions of science in a philosophy of science textbook: none of them specify that science must explain why the current state of the universe is exactly as it is. What science is supposed to do is to predict the outcome of future experiments, and a fundamental multiverse theory would by definition be able to do this for all possible experiments, at least in the probabilistic sense of quantum mechanics.
Moreover, the non-multiverse theories don't actually do a better job of explaining the current state of the universe, anyway. The "standard model", for example, has over 20 adjustable parameters, many of which must be incredibly finely tuned in order for life to exist; what explains this fine tuning? Nothing - it just is. At least with a multiverse theory we have some sort of explanation, namely that all the possible parameter sets are realized in different universes, and we happen to live in this one.
The more one thinks about it, indeed, the more inevitable the multiverse seems. It fits very easily within the probabilistic structure of quantum mechanics. It is a natural extension of the Copernican insight, which has survived every challenge for five centuries now. And it is the only plausible way that our universe could be described by a really beautiful theory, an expectation which is admittedly irrational, but which most theorists deeply believe to be true.
So its time to stop worrying and learn to love the multiverse. It's not going away!