This particular prophecy relates to verse 4, and the particular people referred to are the descendants of the Lehites (Nephi usually includes all of his father’s descendants in these prophecies). It is the future Lehites (termed Lamanites in modern usage of the term) who will be restored to a knowledge of Jesus Christ - through the means of the writings about Jesus that were made by their very ancestors.
Grant Underwood notes the importance of these passages to the early Saints in the last days:
“During the lifetime of the Prophet Joseph Smith, one of the most frequently cited of all Book of Mormon passages was 2 Nephi 30:3-8. In answer to La Roy Sunderland’s Mormonism Exposed, Parley P. Pratt introduced these verses by declaring: ”The Book of Mormon contains many Prophecies, yet future, with names, places and dates, so definite, that a child may understand; indeed, it is one of the peculiar characteristics of the Book of Mormon, that its predictions are plain, simple, definite, literal, positive, and very express, as to the time of their fulfillment. Notice a prediction of Nephi …" (Truth Vindicated 13). Among other events, the passage Pratt referred to foretells the coming forth of the Book of Mormon and its role in restoring the Lehite remnant to a knowledge of their true identity.
“The Indians are the people of the Lord,” wrote W. W. Phelps in his famous series of letters to Oliver Cowdery, “and the hour is nigh when they will come flocking into the kingdom of God, like doves to their windows; yea, as the book of Mormon foretells—they will soon become a white and delightsome people” (“Letter No. 11” 193). Reflecting on the passage describing how the Indians would gain a knowledge of their spiritual heritage through the Book of Mormon, Phelps enthused, “And how much is the joy of our hearts enlarged, when it is known the ’poor Indians,’ are to be raised from their low estate, and miserable condition, by the everlasting gospel; even the fullness of the gospel contained in the Book of Mormon, and other books of God?” (193; emphasis in original).“ (Underwood, Grant. ”Insights from the Early Years." In: Second Nephi, The Doctrinal Structure. Religious Studies Center, Brigham Young University, 1989. p. 333)