3 Nephi is my favorite book in the Book of Mormon because it not only tells the Christmas story in the ancient Americas, but then we have the wonderful chapters telling of Christ's visit there after He was resurrected. These verses are the first things Jesus said to the ancient Americans when He appeared to them. As He did in the old world, He announced who He was and what His mission was. But before He was born, the words were about what He would do. By the time he visited the Americas, He had already done them.
So these words aren't just a "This is what I'm going to do." These words were a declaration, reassurance, proclamation, and explanation of very recent memories for the Savior as He atoned for each of us. If I were among the crowd that heard that declaration, just a few weeks after His sacrifice had occurred, I'm not even sure I could approach my Savior for the awe and wonderment and gratitude that would overwhelm me. Especially to know that He had done so great a thing, and yet didn't forget His other sheep in another part of the world. I love how much the Savior loves each one of His eternal siblings.
(Note: His introduction is very much how our full-time missionaries go about sharing the Gospel. They always introduce themselves as missionaries of the Church, and then they declare what their mission is--that they have a message to share.)
I love the pictures used for this music video, but I'm not the biggest fan of the singing style. So here you go for the story painted:
And here you go for the musical version: