Because Brigham's explanation of the repository in Cumorah is usually edited when it is mentioned at all, here is the entire section straight from the Journal of Discourses, 19:38. I'm using the WordCruncher text. (JD 19, BY Trying to Be Saints • JD 19:38 col. a–b)
There are lots of sources for the Journal of Discourses.
I've broken it up into smaller paragraphs for ease in reading. Original in blue, my comments in red.
I lived right in the country where the plates were found from which the Book of Mormon was translated, and I know a great many things pertaining to that country.
Brigham Young was born in Vermont but his family moved to Smyrna, NY, when he was 3. When he was twelve, the family moved to Aurelius, NY. When he was 14, his mother died and the family moved to Tyrone, NY. When he was 16, he left to be on his own and became an apprentice in Auburn, NY. at age 18, he moved to Port Byron where he joined the Reformed Methodist Church and married Miriam. They moved to Oswego and then to Mendon, when Brigham was 27. He met Heber C. Kimball there. In 1832, Brigham joined the Church in April. His wife Miriam died in September and Brigham and his two daughters moved in the Heber and Vilate Kimball.
I believe I will take the liberty to tell you of another circumstance that will be as marvelous as anything can be. This is an incident in the life of Oliver Cowdery, but he did not take the liberty of telling such things in meeting as I take.
Oliver related his experience privately to at least Brigham Young and David Whitmer, but he could have told others. Publicly, the only thing he wrote that we still have is Letter VII, in which he declares it is a fact that the mile-wide valley west of the Hill Cumorah in New York is the very place where the Jaredites and Nephites fought their final battle to extinction. Some critics reject Letter VII because they say Oliver didn't claim revelation, or he wasn't specific about how he knew the fact of the location of Cumorah. But here, Brigham Young explains how Oliver knew. Oliver had good reason not to discuss this publicly lest he encourage the treasure hunters who were searching the area for the plates.
I tell these things to you, and I have a motive for doing so. I want to carry them to the ears of my brethren and sisters, and to the children also, that they may grow to an understanding of some things that seem to be entirely hidden from the human family.
Brigham explains that he related this account deliberately because he had a specific motive.
Oliver Cowdery went with the Prophet Joseph when he deposited these plates.
The two men went together, according to the principle of two witnesses. We can't tell which plates Brigham was referring to.
Joseph did not translate all of the plates; there was a portion of them sealed, which you can learn from the Book of Doctrine and Covenants.
D&C 28:7 says, "7 For I have given him the keys of the mysteries, and the revelations which are sealed, until I shall appoint unto them another in his stead."
2 Nephi 27:21 explains: "Touch not the things which are sealed, for I will bring them forth in mine own due time; for I will show unto the children of men that I am able to do mine own work."
When Joseph got the plates, the angel instructed him to carry them back to the hill Cumorah, which he did.
We can't tell from this statement which event Brigham referred to, but we can infer from other information that he was referring to the plates of Nephi.
Joseph originally received the abridged plates in September 1827. He received them again in September 1828 (after he had forfeited them after Martin Harris lost the 116 pages). He received the plates of Nephi in June 1829 in Fayette from one of the Three Nephites who had taken the abridged plates to Cumorah. We can tell this because D&C 10 tells Joseph to translate the plates of Nephi, which he did in Fayette.
We also know from Lucy Mack Smith that, before leaving Harmony, Joseph gave the abridged plates to a divine messenger. Joseph did not take those plates back to the hill Cumorah, but the messenger did, as we know from David Whitmer's account of meeting the messenger on the road to Fayette.
Joseph's history explains: "But by the wisdom of God, they [the abridged plates he originally received from Moroni's stone box on Cumorah] remained safe in my hands, until I had accomplished by them what was required at my hand. When, according to arrangements, the messenger called for them, I delivered them up to him; and he has them in his charge until this day, being the second day of May, one thousand eight hundred and thirty-eight." (Joseph Smith—History 1:60)
This account says nothing of taking the plates back to Cumorah. Notice the account says "the messenger," not identifying the messenger by name. Joseph told David that the messenger who had the plates was one of the Three Nephites, which makes sense because this same messenger identified himself to Mary Whitmer as "Brother Nephi," and Nephi was one of the 12 disciples among whom were the Three Nephites. Joseph did deliver up the abridged plates to the messenger before leaving Harmony, so the account in JS-H 1:60 explains what happened to the abridged plates.
Brigham's statement that Joseph took the plates back to the hill Cumorah is a different scenario, but it makes sense when we realize that the messenger gave Joseph the plates of Nephi in Fayette. It makes sense that the messenger would tell Joseph to return these plates to Cumorah because Joseph was going to show these plates to the 8 witnesses in Palmyra, as his mother explained. Recall that none of the 8 witnesses said any part of the plates was sealed. Only the abridged plates had a sealed portion.
Naturally, once in Palmyra, Joseph could easily return the plates to the repository in Cumorah.
Oliver says that when Joseph and Oliver went there [to the Hill Cumorah], the hill opened, and they walked into a cave, in which there was a large and spacious room.
This can create a strange image if we imagine it like an elevator door opening into a cave. But if instead we understand it as a door on the ceiling that could be raised, or opened up, so that a person could walk in on steps or a ladder, then the explanation makes sense. Such an opening is easily concealed on the outside with brush, leaves, dirt, etc. The term "cave" does not mean only a natural cave. It can also apply to a man-made chamber. The photo of the room on Cumorah shows a room made up of stone walls with an opening in the top. Wilford Woodruff said the room was about 16x16, which would qualify as "large and spacious" by the standards of typical rooms in houses at that time.
He says he did not think, at the time, whether they had the light of the sun or artificial light; but that it was just as light as day.
This makes sense when we realize the opening was in the ceiling. The sun would shine down into the room. But there could have also been artificial lights, such as the Jaredite stones that gave off light.
They laid the plates on a table; it was a large table that stood in the room. Under this table there was a pile of plates as much as two feet high, and there were altogether in this room more plates than probably many wagon loads; they were piled up in the corners and along the walls.
This quantity of plates is consistent with what Mormon wrote: "having been commanded of the Lord that I should not suffer the records which had been handed down by our fathers, which were sacred, to fall into the hands of the Lamanites, (for the Lamanites would destroy them) therefore I made this record out of the plates of Nephi, and hid up in the hill Cumorah all the records which had been entrusted to me by the hand of the Lord, save it were these few plates which I gave unto my son Moroni." (Mormon 6:6)
One frequent question is why would Mormon have chosen the Hill Cumorah as the place of the final stand. It is strategic in the sense of being the largest hill in the area, with access to water, surrounded by a series of drumlins that served as natural fortifications to deter advancing forces. But why this particular hill?
We know the same hill was called Ramah by the Jaredites. Mormon likely knew the Jaredites had fortified the hill with underground chambers that would provide an accessible, effective hiding place for the repository of Nephite records. Such chambers would be difficult to replicate elsewhere, especially on a short time frame.
Another important aspect of Brigham's statement is that there were "more plates than probably many wagon loads." He doesn't specify how big such wagons would be, but why that measure?
The statement suggests that Brigham knew Joseph, Oliver, and likely others moved the plates before leaving Palmyra, and used wagons to do it. (A logical site would be the hill Shim which was the repository from which Mormon took the plates in the first place.) Apparently neither Joseph nor Oliver told Brigham how many wagon loads, but given the stacks of plates, Brigham assumed it required "many" loads.
Summary: Brigham Young recognized that what was common knowledge among those who personally knew Joseph and Oliver could easily be forgotten and lost if it was not made a matter of record. He specifically invoked that risk when he related this account of the repository in the New York Cumorah. The account was corroborated by several others, many collected in the paper "Cumorah's Cave."
In the early 1920s, LDS scholars began adopting the Mesoamerican/Two-Cumorahs (M2C) theory invented by L.E. Hills. Obviously, Brigham Young's statement directly contradicted M2C. Thus, LDS scholars who embrace M2C have successfully de-correlated the New York Cumorah to the point where, just as Brigham feared, the knowledge has been "forgotten and lost."
Fortunately, Brigham's sermon was recorded and published, and now it is available everywhere on the Internet.