The next paragraph of the talk in OP gives context:
Adam held the keys of the First Presidency and stood directly after the Savior in authority (see Teachings, p. 168). He received those keys in the Creation, according to the Prophet Joseph Smith, who added, “Christ is the Great High Priest; Adam next” (Teachings, pp. 157–58).
The referenced article can be referenced as well.
In regards to priesthood keys1:
What are priesthood keys?
Priesthood keys are the right to preside over and direct the Church within a jurisdiction. Keys usually apply to a geographic area, like a ward, stake, or mission. They also usually include authority over specific ordinances and activities (for instance, baptism, the sacrament, missionary work, and temple work).
Where do priesthood keys come from?
All priesthood keys ultimately come from Heavenly Father through His Son, Jesus Christ. Joseph Smith taught, “The keys [of the priesthood] have to be brought from heaven whenever the Gospel is sent” (Teachings of Presidents of the Church: Joseph Smith , 104). The keys necessary for governing the Church in the latter days were restored to Joseph Smith by heavenly beings—for instance, John the Baptist, Peter, James, John, Moses, Elias, and Elijah (see Joseph Smith—History 1:72; D&C 27:12; D&C 110:11–12). These keys have been passed on to his successors.
During each dispensation, "the Lord has at least one authorized servant on the earth who bears the holy priesthood and the keys, and who has a divine commission to dispense the gospel to the inhabitants of the earth." Adam was the authorized servant during the first dispensation. All dispensations PDF
How could the Church of Jesus Christ be well established even before Jesus incarnated or before He was even born?
The Church of Jesus Christ of Latter-day Saints believes the earth needed a restoration of apostolic authority, which is referred to as the priesthood and has been on the earth during varying times/dispensations. At times between dispensations there were times of apostasy. This is a recognized pattern LDS believe2:
The Lord calls prophets to teach His children the plan of salvation. Adam was the first prophet, followed by others, such as Noah, Abraham, and Moses. Time and again throughout the history of the world, the people eventually rejected the prophets’ message. When this happened, prophets, ordinances, and priesthood authority were taken away and the people lived in spiritual darkness. These periods of darkness are referred to as periods of apostasy. Because of His mercy, in time the Lord called and sent new prophets to restore His gospel. Understanding this pattern of teaching, rejection, and restoration prepares a person to recognize the Lord’s hand in the latter-day Restoration through the Prophet Joseph Smith, who opened the dispensation of the fulness of times.
All emphasis mine
1 New Era: Priesthood Keys, True to the Faith: Priesthood, Keys of the Priesthood by then Elder Russell M Nelson
2 Missionary Preparation Manual: Prophets and Apostasy