Chapter 21 of 1 Nephi is only slightly different from Isaiah 49. Perhaps even more than Isaiah 48, it is clear in this first verse why Nephi would liken this chapter to his people.
First, it is a statement to all of the house of Israel. This inclusive statement would easily have been seen as including the house of Israel in the New World. While Isaiah would have originally been speaking to the scattered ten tribes, the Nephites certainly saw themselves as “broken off and … driven out because of the wickedness … of my people.”
Nephi recorded that his father, Lehi, had spoken of the allegory of the olive tree in 1 Nephi 10:12. In that verse, he specifically mentioned that the gentiles would be like branches broken off and scattered. Thus, Isaiah’s message to an Israel that had been broken off and driven out would have been seen as an echo of the allegory of the olive tree.
Even more specifically, Isaiah speaks to those who are on the isles (or more accurately, isles of the sea—as in verse 8 in this same chapter), which Nephi repeatedly associates with his family in the New World. Particularly, since those who are to listen to the Lord are those in the isles who are, Isaiah says, “people from far.”
For Nephi, it would have seemed that this entire verse pointed directly at his people.