This is such an amazing concept that goes way deeper than we could possibly cover in a short blog post. This has all to do what what we do as missionaries, and the Second Coming of Christ. So let's break this one down a little bit as well.
First, "We believe in the literal gathering of Israel and in the restoration of the Ten Tribes." As we read in the Bible, the tribes of Israel were once all together, but were then split into the North and South, the North included the Ten Tribes, and the South included two tribes, Judah and Benjamin. History shows us that the prophecies in the Bible were fulfilled, and Ephraim, or the Ten Tribes, and Judah, the Two Tribes, were scattered and destroyed. Later, we also understand from history that Judah was allowed back into Jerusalem and they are the major tribe that is in Jerusalem today, but that still leaves the Ten Tribes. They got scattered all across the world, and it is our job to go and find those people, and to bring them back, or in other words restore the Ten Tribes.
Second, "We believe that Zion (the New Jerusalem) will be built upon the American continent." This one comes due to modern revelation. Zion, or the New Jerusalem, will be built here in America. The Lord has said that Zion will be in Independence, Missouri.
Third, "We believe that Christ will reign personally upon the earth." Christ is going to be the guy in charge! That will be a great day. The Lord himself will reign. His presence will be felt and all the world will praise him.
Fourth, "We believe that the earth will be renewed and receive its paradisaical glory." The earth is going to be awesome! Everything will be perfect. It's like the earth getting resurrected, and so just like us it will be perfect. That will be the day.
All of this will be taking place before, during, and after the millennium. It is going to be a great and dreadful day. Hopefully great for many rather than a few.