I understand the three levels of Heaven. If non-Mormons cannot enter the celestial kingdom, does that mean that it was empty of spirits before The Book of Mormon had been translated and the LDS church started? It hard to believe an eternal heaven before us being vacant until then.
The Church of Jesus Christ of Latter-day Saints claims to be a restoration of the New Testament Church founded by Christ and guided by the Apostles. In addition to that, we claim that the Gospel of Christ was preached beginning with Adam. We know Moses tried to get the Israelites to live the Gospel of Christ, but due to their hardheartedness, rejected it and ended up with the Law of Moses, which was designed to soften their hearts, so they would accept and live the Gospel.
We also believe that those who would have accepted the Gospel of Christ, but didn’t get the chance, also go there. This is explained in the Doctrine and Covenants 137:5-10
5 I saw Father Adam and Abraham; and my father and my mother; my brother Alvin, that has long since slept;
6 And marveled how it was that he had obtained an inheritance in that kingdom, seeing that he had departed this life before the Lord had set his hand to gather Israel the second time, and had not been baptized for the remission of sins.
7 Thus came the voice of the Lord unto me, saying: All who have died without a knowledge of this gospel, who would have received it if they had been permitted to tarry, shall be heirs of the celestial kingdom of God;
8 Also all that shall die henceforth without a knowledge of it, who would have received it with all their hearts, shall be heirs of that kingdom;
9 For I, the Lord, will judge all men according to their works, according to the desire of their hearts.
10 And I also beheld that all children who die before they arrive at the years of accountability are saved in the celestial kingdom of heaven.
The idea that only Mormons will go to the Celestial Kingdom is clearly not correct. That being said, the Mormons currently claim to be the ones God has given the authority to perform the necessary ordinances of the Gospel. Therefore those desiring to gain the highest degree of the Celestial Kingdom will need to meet the exceptions outlined in the above scripture or join The Church of Jesus Christ of Latter-day Saints and live their faith to the fullest extent they can.
Please keep in mind however, that the designation of kingdoms is not done until the final judgment is made. Until then, those that are righteous go to Paradise awaiting the final judgment.