I come from an LDS family, so I grew up with a set of traditions surrounding Joseph Smith and the Book of Mormon. There are numerous Mormon sects, each with their own traditions, and not all of them in agreement with the others. In the mainstream LDS group, people generally thought that Joseph had something called the "urim and thummim," which was a kind of loupe or spectacles that the prophet would gaze through in order to divine the writing on the gold plates. He would read along in this manner, dictating to his scribe. The urim and thummim is based in old Testament lore, and Joseph claimed to have found one of these divining objects along with the gold plates, an armor breastplate, and a sword, all buried together in the Cumorah hill.
Smith got through 116 pages of manuscript and then loaned these out to a friend so that the friend could show the work to some interested parties. These 116 pages were subsequently lost, at which time Joseph Smith said that a divine messenger came to retrieve the gold plates along with the urim and thummim. The story goes that, after some time passed, Joseph again found favor with God and was reinstated as translator, only this time without the aid of the urim and thummin. Instead, Smith used his seer stone.
Whether or not this is a reasonable explanation for a religious text, it is not inconsistent with the American frontier folk magic of the early nineteenth century. Growing up with these traditions, most Latter-Day Saints have no problem with them. Because many of them didn't know about the seer stone until very recently (the church published some photos, which you can probably find online), it is a challenge for them to believe that translation via seer-stone is a "reasonable explanation," as you say. It doesn't exactly square with the urim and thummim version of the story that they had originally. I'm sure this is all absurd nonsense to most outsiders, although the LDS church is maintaining its congregation and perhaps even growing it slightly, in spite of whatever perceived absurdity exists within its history. Mormons are by no means particularly stupid or credulous, and generally the church doesn't bank on its history when looking for converts.
Reasonable or not, I find Mormon history quite fascinating. Well, that is, up until these recent times in which we now find the largest sect of Mormonism little more than a gigantic corporate franchise. Speaking of America, Eric Hoffer once said that "what starts out here as a mass movement ends up as a racket, a cult, or a corporation." Indeed. Nevertheless, you'll find the Mormons to be an excellent lot in spite of their sappy, corporate church.