The Mormons point to 2 phrases in chapter 10 as supporting 2 unusual points of their doctrine.
First, in verse 16 Jesus says "I also have other sheep that are not in this fold; I must also bring them, and they will hear my voice, and there will be one fold, one shepherd."
According to the Book of Mormon, Jesus came to the Americas during the day or two that passed between his crucifixion and the empty tomb. The Church of Jesus Christ of Latter-Day Saints says that in verse 16 Jesus was referring to the people of the Americas who he would speak to during that time.
Second, LDS doctrine says that we are all intended in the afterlife to be literal gods, each with our own universe, just like the man we know as Yahweh is the god of this universe and Jesus and Lucifer are his sons. There are many other universes, each with their own god and his wives and children.
So the LDS say that Jesus was referring to this reality about the universe when he quoted the phrase from Psalm 82:6, "You are all gods, and you are all sons of the most high."
As a staunch backslidden non-practicing Unitarian-Universalist myself, I claim the heritage of
all religions, as long as it doesn't hurt anybody. The Mormon version might as well be true if anybody wants it to be.