Physicist and writer Brian Greene does a fine job of explaining the concept of the multiverse in a column at The Daily Beast.
In the piece he quotes Carl Sagan as saying that extraordinary claims require extraordinary evidence and then tacitly acknowledges that there's not much evidence for the multiverse theory, so we're left to wonder why it has enjoyed so much popularity among some cosmologists.
Perhaps one reason is that the universe is comprised of forces and constants whose values are calibrated with unimaginably exact mathematical precision. If any of dozens of forces, like gravity, for instance, deviated in their strength from the tiniest amounts - one part in 10^40 in the case of gravity - the universe could not exist, or if it did it would not be the sort of place where living things could emerge.
It's mind-bendingly improbable that such precision would have emerged by sheer chance and there are thus only two viable explanations for it. Either the universe is the product of an intelligent engineering process or there are so many different universes, an infinite number, that one like ours would have to exist. Just as the probability of a blind-folded shooter hitting a postage stamp half a mile away is increased as the number of bullets fired increases, so, too, the chance of a universe like ours appearing increases as the number of different universes that are produced increases.
It seems odd that scientists would posit an explanation which requires the existence of so many entities for which there's so very little evidence, but consider that the only viable alternative is that the universe is the creation of an intentional agent, a God, and it's easier to understand why they do so. It is, at least for some of them, an act of metaphysical desperation.
Anyway, it would be good to read Greene's article. It's written by a physicist who's sympathetic to the multiverse theory and, like much of his work, it's very lucid and accessible to the layman.