1 Nephi 15:15-16

In verse 15, Nephi is stating that when their seed, his and his brothers', receive the gospel again, would they rejoice? Yes! They will be nourished from the true vine. They will be a part of the true fold of God. Because of their conversions, they will be remembered again of God and become a natural branch of the olive tree and be grafted into the true olive tree again. Nephi says that Lehi said that this would not happen until after their seed is scattered by the Gentiles. Then it will be the Gentiles that will bring the gospel to the remnant of the seed of Nephi and his brothers.

Here is what the Book of Mormon Institute Manual says about the "true vine" and the work:

Of the great latter-day work among the Lamanites, President Spencer W. Kimball stated:
“Of immense importance to this work of gathering the scattered branches of the house of Israel is the work of carrying the blessings of the restored gospel of Jesus Christ to the Lamanites, for the Lord’s work in these latter days can in no wise be complete until these children of great promise are brought back into the fold. The Lord said through his prophet Lehi, ‘Behold, I say unto you, Yea; they shall be remembered again among the house of Israel; they shall be grafted in, being a natural branch of the olive-tree, into the true olive-tree.’ (1 Ne. 15:16.) We are witnesses to these events. . . .
“. . . For a thousand years after the closing of the Book of Mormon record, these people wandered in spiritual darkness and were scattered upon the American continents and the isles of the sea. They lost their written language, their high culture, and, worst of all, their knowledge of the living God and his work. Faith was replaced by fear, rich language by crippled dialects, and an understanding of God and his ways by idolatry. . . . Only the most brazen soul could fail to weep when contemplating the fall of this people, and yet it was the decree of the Lord that the Lamanites should be preserved in the land, that this remnant of Joseph should again come into their promised inheritance. . . .
“. . . The Lord’s promises with regard to the Lamanites began to be fulfilled with the coming forth of the Book of Mormon in this dispensation (see Eth. 4:17). . . .
“Truly our paths have met once more—we a mixed remnant of Israel, principally Ephraim, even referred to as gentiles, now come forth out of captivity (see, e.g., 1 Ne. 13:19, 39), . . . now only through the grace of Almighty God restored to the blessings of the gospel, that we in turn might be a blessing to the nations of the earth; and the Lamanites, also a people of disobedience now returned to the fold. . . .
“The Lamanites must rise again in dignity and strength to fully join their brethren and sisters of the household of God in carrying forth his work in preparation for that day when the Lord Jesus Christ will return to lead his people” (“Our Paths Have Met Again,” Ensign, Dec. 1975, pp. 4–5, 7).