How was the translation of the Book of Mormon shown through Joseph Smith's seer stones and hat? The Wikipedia article on the golden plates says that Joseph Smith translated the golden plates into the Book of Mormon with the help of "seer stones". Initially, he is said to have used two stones set in frames as glasses to see the new translation. Later, he switched to a different method with one seer stone inside a hat, but I'm confused as to how it was supposed to help in translation.


  After the loss of the first 116 manuscript pages, Smith translated with a single seer stone, which some sources say he had previously used in treasure-seeking. Smith placed the stone in a hat, buried his face in it to eliminate all outside light, and peered into the stone to see the words of the translation. […] Smith's translation did not require the use of the plates themselves. Though Smith himself said very little about the translation process, his friends and family said that as he looked into the stone, the written translation of the ancient script appeared to him in English.




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I'm confused about this process. Obviously, there is some sort of divine power assumed to be going on here, but I’m trying to understand how the miracle was said to have manifested, specifically where these letters appeared.

Was the stone projecting the words into the darkness? Were the words appearing upon the face of the stone, like the popular depiction of a crystal ball? Was the stone moving around and forming letters? How was the translation of the Book of Mormon shown through Joseph Smith's seer stones and hat?

In other words, how does hat + stone = words?
From LDS.org we read:

The other instrument, which Joseph Smith discovered in the ground years before he retrieved the gold plates, was a small oval stone, or “seer stone.” As a young man during the 1820s, Joseph Smith, like others in his day, used a seer stone to look for lost objects and buried treasure. As Joseph grew to understand his prophetic calling, he learned that he could use this stone for the higher purpose of translating scripture.
Apparently for convenience, Joseph often translated with the single seer stone rather than the two stones bound together to form the interpreters. These two instruments—the interpreters and the seer stone—were apparently interchangeable and worked in much the same way such that, in the course of time, Joseph Smith and his associates often used the term “Urim and Thummim” to refer to the single stone as well as the interpreters. In ancient times, Israelite priests used the Urim and Thummim to assist in receiving divine communications. Although commentators differ on the nature of the instrument, several ancient sources state that the instrument involved stones that lit up or were divinely illumined. Latter-day Saints later understood the term “Urim and Thummim” to refer exclusively to the interpreters. Joseph Smith and others, however, seem to have understood the term more as a descriptive category of instruments for obtaining divine revelations and less as the name of a specific instrument.

Concerning the mechanics of how this took place, the same source reports:

the scribes and others who observed the translation left numerous accounts that give insight into the process. Some accounts indicate that Joseph studied the characters on the plates. Most of the accounts speak of Joseph’s use of the Urim and Thummim (either the interpreters or the seer stone), and many accounts refer to his use of a single stone. According to these accounts, Joseph placed either the interpreters or the seer stone in a hat, pressed his face into the hat to block out extraneous light, and read aloud the English words that appeared on the instrument. The process as described brings to mind a passage from the Book of Mormon that speaks of God preparing “a stone, which shall shine forth in darkness unto light.”
The scribes who assisted with the translation unquestionably believed that Joseph translated by divine power. Joseph’s wife Emma explained that she “frequently wrote day after day” at a small table in their house in Harmony, Pennsylvania. She described Joseph “sitting with his face buried in his hat, with the stone in it, and dictating hour after hour with nothing between us.” According to Emma, the plates “often lay on the table without any attempt at concealment, wrapped in a small linen table cloth.” When asked if Joseph had dictated from the Bible or from a manuscript he had prepared earlier, Emma flatly denied those possibilities: “He had neither manuscript nor book to read from.” Emma told her son Joseph Smith III, “The Book of Mormon is of divine authenticity—I have not the slightest doubt of it. I am satisfied that no man could have dictated the writing of the manuscripts unless he was inspired; for, when acting as his scribe, your father would dictate to me for hour after hour; and when returning after meals, or after interruptions, he would at once begin where he had left off, without either seeing the manuscript or having any portion of it read to him.”
Another scribe, Martin Harris sat across the table from Joseph Smith and wrote down the words Joseph dictated. Harris later related that as Joseph used the seer stone to translate, sentences appeared. Joseph read those sentences aloud, and after penning the words, Harris would say, “Written.” An associate who interviewed Harris recorded him saying that Joseph “possessed a seer stone, by which he was enabled to translate as well as from the Urim and Thummim, and for convenience he then used the seer stone.”

This is all that we know.  If you're looking for the physics behind these events, we don't have that to give you any more than we can the process of resurrection.