As part of the transformation, the three are caught up into heaven. This may have been a revelation given to them to help them understand the complete nature of their long ministry on the earth. There is no explanation as to why this heavenly experience was given to the three but not the other nine.
The emphasis in verse 15 is on the concept of transfiguration. At least the concept of transfiguration comes from the language describing the experience of the Savior in his mortal ministry:
Mark 9:2-3
2 And after six days Jesus taketh with him Peter, and James, and John, and leadeth them up into an high mountain apart by themselves: and he was transfigured before them.
3 And his raiment became shining, exceeding white as snow; so as no fuller on earth can white them.
The transfiguration of Jesus was a change that altered the way he appeared, and that appearance was from mortal into a being of light. This is the imagery of the shining raiment, and common description of heavenly beings and messengers.
The need for a transfiguration of some sort to “behold the things of God” comes from the Old Testament passage:
Exodus 33:20
20 And he said, Thou canst not see my face: for there shall no man see me, and live.
That which is mortal and unprotected cannot withstand the presence of God, or the “things” of God, and therefore the human must undergo some type of transfiguration to be able to survive that experience. In the case of the three Nephites, that transfiguration became permanent in some form so that their moral body was no longer subject to the pangs of mortality.
Reference: The language describing this experience is borrowed from Paul:
2 Corinthians 12:2-4
2 I knew a man in Christ above fourteen years ago, (whether in the body, I cannot tell; or whether out of the body, I cannot tell: God knoweth;) such an one caught up to the third heaven.
3 And I knew such a man, (whether in the body, or out of the body, I cannot tell: God knoweth;)
4 How that he was caught up into paradise, and heard unspeakable words, which it is not lawful for a man to utter.
The similarity of experience is certain. The process of being “caught up” in some form to heaven and experiencing that realm would certainly lead to ineffable experience. What happened was independent of the New Testament passage, but obviously the language used to describe that similar experience is influenced by the Pauline text.