This is actually a pretty strange topic because the Book of Mormon makes fantastic claims that were not written based on historical evidence or even freewheeling extrapolation about actual events (like various "Ancient Aliens" theories), but instead based on a set of theoretical golden plates written by ancient Native Americans translated through supposed revelation from God to a man who wasn't very educated in Native American history. It seems fair to me that the book absolutely fails to depict any actual events in Mesoamerica, or any other part of America for that matter (some LDS apologist claim that the book takes place in the Great Lakes region, for example).
The book is often very vague, which makes it easy for believers to attach themselves to very specific ideas, only to pivot to new positions when counter evidence arrives. For example, you should read this article, written by an LDS apologist, about the Book of Mormon's claim that ancient Native Americans rode horses, something that European settlers/conquerors introduced to the Americas far after the original record was supposedly written: http://www.fairmormon.org/perspectives/publications/horses-in-the-book-of-mormon
Another similar example: before 2006 the introduction claimed that the cultures described in the Book of Mormon "were destroyed, except the Lamanites, and they are the principal ancestors of the American Indians." After DNA evidence was discovered that the Native Americans are descended from Asian migrants and not Jewish pilgrims the introduction was changed to say that "the Lamanites 'are among the ancestors of the American Indians.'"
The book also discusses some awesome and incredible events that supposedly occurred to the Nephite (the culture that the book focuses on) and Lamanite peoples, such as the great destruction that happened in the Americas in the wake of Christ's death at Jerusalem. The number of cities that were described as being burned to the ground, buried beneath the earth, and sunk under the sea in the period of three days is staggering (Zarahemla, Moroni, Moronihah, Gilgal, Onihah, Mocum, Jerusalem (presumably a city in America called Jerusalem?), Gadiandi, Gadiomnah, Jacob, Gimgimno, Jacobugath, Laman, Josh, Gad, Kishkumen, and perhaps more). Such an event would have been one of the most cataclysmic events in human history, but the existence of any of these cities is completely unverified.
Similarly, the Book of Mormon also describes a civil war that took place at ca 600 BCE when an entire civilization of at least 2 million men (perhaps 4 million depending on your interpretation) (and presumably more women and children) wiped itself out to literally the last man, when the king of one side of the conflict killed the leader of the opposing faction in a one-on-one duel. This would represent perhaps the most bloody war before the common era (someone with more historical chops might be able to correct me on that, but I'm fairly sure that's true), and as far as I have seen no evidence of such an event exists.
Source: https://www.lds.org/scriptures/bofm/3-ne/9.9
https://www.lds.org/scriptures/bofm/ether/15?lang=eng
Essentially, the book itself tells its readers that the way to believe that the book is true is to have faith and to pray about it. As the evidence and our modern understanding of Native American history stands this seems to be the only method that a person could use to believe in the literal truthfulness of the Book of Mormon. Does the book function as a religious text? That is a separate, and maybe an even more interesting, discussion.
Edit: I should probably disclose the fact that I was raised Mormon and left the church as an adult, but I did my best to remain impartial and to cite my sources. If someone has an issue with the way I delivered my thoughts please let me know!