How was the translation of the Book of Mormon shown through Joseph Smith's seer stones and hat?

From the most substantial witness accounts and traditional uses of seer stones, it seems that the function of the seer stone was to be an object of mental/spiritual focus to help Joseph Smith attain a visionary state. The hat also assisted with that by blocking out distraction and disrupting the usual visual processes. The parchment appeared in vision (and the translation on the parchment), similar to how we see a dream, without the aid of the physical eye or physical light. One associate of JS said he could see the translation as well with his eyes closed as open. Compare biblical accounts of books (called rolls or scrolls anciently) read in vision by John in Revelations and by Ezekiel. Search for the article, "Seers and Stones" for the historical evidence. The vision is the "spiritual light" that shone in the darkness.

In response Thunderforge's apt comment below, here are three quotes (with context), lifted from the mentioned article (which, in full disclosure, I authored in Interpreter):

In a letter to another Methodist minister dated October 24, 1831, Booth notes the similarity between Joseph Smith’s visions of celestial beings and his translation of the Book of Mormon:

Smith is the only person at present, to my knowledge, who pretends to hold converse with the inhabitants of the celestial world. It seems from his statements, that he can have access to them, when and where he pleases. He does not pretend that he sees them with his natural, but with his spiritual, eyes; and he says he can see them as well with his eyes shut, as with them open. So also in translating. — The subject stands before his eyes in print, but it matters not whether his eyes are open or shut; he can see as well one way as the other…These treasures were discovered several years since, by the means of the dark glass, the same with which Smith says he translated the most of the Book of Mormon.”

The “dark glass” that Joseph Smith used to translate “most of the Book of Mormon” in Booth’s account accords with the stone of “rather a dark color” mentioned by Emma Smith and the “dark colored, opaque stone” mentioned by David Whitmer.

There is a similar description of Joseph Smith closing is eyes while also translating the Book of Abraham with seer stone. Note that the biblical definition of "seer" is a "see-er" of visions. Again, a quote and context from the article:

Wilford Woodruff, in reporting the use of the “urim and thummim” to translate the book, also called Joseph Smith a seer. John Whitmer’s history of the Church also portrays Joseph Smith as translating the Book of Abraham in the capacity of seer: “Joseph the Seer saw these Record[s] and by the revelation of Jesus Christ could translate these records.”

The only other account of the translation of the Book of Abraham from a potential witness is from Lucy Smith, although it is secondhand at best. A group of Quakers who visited Lucy Smith reported in 1846 that she told them that

when Joseph was reading the papyrus, he closed his eyes, and held a hat over his face, and that the revelation came to him; and where the papyrus was torn, he could read the parts that were destroyed equally as well as those that were there; and that scribes sat by him writing, as he expounded.

And back to JS translating the Book of Mormon with a hat pulled "closely around his face," which, in a lamp-lit room, would make the interior of the hat pretty dark...

The stone disappears in the darkness and anything that is seen must be seen, in David Whitmer’s words, by “spiritual light.” According to a report of an interview by James H. Hart in 1884, David Whitmer described the disappearing act of Joseph Smith’s seer stone as it was replaced by a vision of sacred text:

The way it was done was thus: Joseph would place the seer-stone in a deep hat, and placing his face close to it, would see, not the stone, but what appeared like an oblong piece of parchment, on which the hieroglyphics would appear, and also the translation in the English language.

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.