Christ reminds Elisha that He has set all these events in motion in order to remind the people of the islands that:
1) He is Lord of all nations and King in all lands and of all people
2) they are His children and one of His chosen people
3) He visited them in ancient times
4) He made covenants with them that He intends to keep
5) they will be gathered back into the House of Israel and receive their blessings
6) the island kingdom is given to them for their inheritance
7) the Aklatan is the catalyst for this chain of events
He tells Elisha that at the appointed time Oleeha will return to aid in translating the record, which speaks concerning the time when He came unto the island people. By His power, Christ caused it to be preserved in the cave and Elisha to discover it that it might come forth among the people and be a guide for them. The Aklatan will also be proof (He uses the word "proof" and not just "evidence") of two things: Elisha's calling and the Holy Spirit's power to witness of truth.
This promise from Jesus is key for anyone with hangups over Elisha. It's also analogous to Joseph Smith and the Book of Mormon. In the early days of the Church of Jesus Christ of Latter-day Saints, the Book of Mormon was not read or studied as thoroughly as it is today; at first, it was mainly just a physical manifestation and witness of Joseph's prophetic calling. Even today, LDS missionaries ask everyone to read the Book of Mormon and receive a witness from the Spirit of it's truth. Then, the logical progression follows that if the Book of Mormon contains the word of God, then Joseph was a prophet of God who restored Gospel knowledge, priesthood keys, Christ's church, etc. in the latter days. The promise is the same, then, for Elisha: seeking and receiving a witness from the Holy Spirit of the truths found in the Aklatan logically leads to a witness of his divinely issued calling to the people of the islands.
Elisha is commanded to write all of Jesus' words down and promised they will stay in his memory until he has done so. At the end of the chapter, after Christ departs, Elisha relates that this happened just as Christ said it would; he remembered the whole visitation with perfect clarity until he wrote it down, and only then the memory began to fade. This reminded me of the story behind D&C 132. Historically, we can deduce that Joseph had received the revelation on celestial marriage as early as 1831, but it wasn't written down until 1842 at the insistence of his brother, Hyrum. Joseph declined using the Urim and Thummim, saying that he knew the revelation perfectly from beginning to end, even though he'd kept it in his mind for some 11 years. After recording it and making a copy, Joseph said that much more could've been written, but what was recorded was sufficient at that time. He even allowed his wife, Emma, to destroy the original, claiming he could recite it word for word at any time if necessary.
Wilford Woodruff also spoke of this unique spiritual gift (h/t Bruce):
Whenever I hear Joseph Smith preach, teach, or prophesy, I always felt it my duty to write it; I felt uneasy and could not eat, drink, or sleep until I did write; and my mind has been so exercised upon this subject that when I hear Joseph Smith teach and had no pencil or paper, I would go home and sit down and write the whole sermon, almost word for word and sentence by sentence as it was delivered, and when I had written it, it was taken from me, and I remembered it no more. This was the gift of God to me.Christ's final command is to spread these words among the people, and whoever obeys these commands will be saved in the last day.