Physicist and writer Brian Greene does a fine job of explaining the concept of the multiverse in a column at The Daily Beast which appeared some years ago.
In the piece he quotes the late Carl Sagan reminding us that in science extraordinary claims must be supported by 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 our 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 10^60 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 as a consequence of sheer probability.
Just as the probability of a blind-folded rifleman hitting a postage stamp half a mile away is increased as the number of bullets fired increases, so, too, the chance of a universe as improbable as ours increases as the number of different universes that exist increases toward infinity.
It seems odd that scientists would posit an explanation which requires the existence of so many entities for which there's so 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.
As physicist Bernard Carr once put it, "If you don't want God you better have a multiverse." They're the only two live options.
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.