They’re quoting from John 10:16, where Jesus says, "And other sheep I have, which are not of this fold: them also I must bring, and they shall hear my voice; and there shall be one fold, and one shepherd."
Mormons claim that these words of Christ were fulfilled when, after his death and resurrection, he visited the Americas to establish a church among the "Nephites." These "lost sheep" were supposedly the descendants of Hebrews who had fled Jerusalem and journeyed to America at the time of Jeremiah. The Book of Mormon purports to be the religious and historical records of these ancient "Christians" (as the Book of Mormon records they called themselves, even before the coming of Jesus), written and preserved by American counterparts of Hebrew prophets.
Under the rubric, "God is no respecter of persons" (Acts 10:34), the Mormon church finds its rationale for the above interpretation and argues that God deals fairly with all his children. Since Christ was sent to Palestine to teach and establish a church among the Jews there, it was only right that he come also to the Americas and establish a church there as well. As so often is the case with Mormon arguments, if they prove anything, they prove too much. The "God is no respecter of persons" principle, both in Acts 10 and elsewhere (Rom 2:11), is applied in a specifically Jewish-Gentile context, showing that God treats all groups fairly. Thus a Mormon could not limit the "no respecter of persons" principle to just those who are members of the house of Israel (Old World or New World Israel). It includes all men, everywhere. If, because of this principle, he had to come to North America and start a church to be fair to the people there, then he would have to visit every continent, and all the peoples on those continents, and start new, independent churches everywhere, in order to show that he is no respecter of persons.
The logical alternative is to say that God shows his grace by starting a single, unified church—somewhere—and that this church is to receive all men, Jew and Gentile. By having a single church, when it expands, men will have no doubt about which one they are to join. Until the time that this church reaches them, they will be judged based on whatever knowledge of God they have—however much or little that may be—and their willingness to follow his truth if they had known what it was.
In reality, the "other sheep" Jesus mentions are the righteous Gentiles, who did not belong to the "fold" of God’s chosen people, Israel, but who would respond to the gospel when preached to them. While Christ’s earthly ministry served the Jewish people almost exclusively, his great commission to the apostles before his ascension sent them into all the world to preach, baptize and thus unite his believers in one fold (Mt 27:19). Because "he that heareth you heareth me" (Lk 10:16), to hear the gospel from the lips of his disciples is to hear Jesus himself.
The understanding of the "other sheep" as the Gentiles who would come to believe in Christ is the natural understanding of the passage. Mormons sometimes ask Christians, "If the ‘other sheep’ weren’t in the New World then who were they?"
A Christian often will be perplexed at the fact the question was asked at all and respond, "Well, they’re the Gentile Christians, of course. How could anyone think the text suggests otherwise?" The New Testament has a running theme of how salvation comes from the Jews to the Gentiles. It appears across multiple books, in all of the gospels and most of the epistles. Jesus’ statement about gathering other sheep in the future is simply one more instance of the gospels dealing with this theme.
The fact that Mormons often do not spot the obvious, face-value interpretation of the text reveals how little Mormons have been exposed to the historic understanding of the passage and how little they have been encouraged to think through its rationale. They have not tried to understand the New Testament as a whole, integrating and understanding its individual passages with other passages and with the general historical backdrop. Instead, they have had the interpretations of certain alleged proof texts force-fed to them in a way that keeps them from knowing of the existence of other, more plausible interpretations.