Yay, a question where I actually have some level of cursory expertise!

TL;DR at the end. Sorry for typos, didn't proof before submitting.

To begin understanding how Mormonism was allowed to flourish requires a brief understanding of the Protestant melting pot that was 19th-century America. Joseph Smith (founder of Mormonism) was born into what was known as the "burned-over district" during the Second Great Awakening, a veritable hotbed of theological and religious development. Other notable religions cropping up from the same area are Seventh-Day Adventists, Disciples of Christ, Millerites, etc. {The Making of a Prophet, Vogel}

Campbellite theology was making its way into popular religious culture and influencing theological discussions with public debates with Campbellite leaders like Alexander and Barton Stone. Young Joseph Smith (1805-44) grew up likely attending local revival meetings.

For a boy of a family of 12 with a vagabond father and no sustainable income or regular education opportunities, you can imagine his eyes bulging as he saw the collection plates passed around these congregations after sermons. A relevant piece of Smith's personality should be noted here; Smith was a charismatic man. Charisma has its benefits and weaknesses. He was polarizing, non-committal, and had a malleable set of beliefs and convictions which could quickly adapt to setting and audience. One can see how such a personality would do well as a magnetic religious leader.

The set of factors which caused Smith to claim he had an ancient record of Native American Jews is a rabbit hole too deep to dive into with this question, but suffice it to say, eventually he claimed to have such a record written on Gold Plates, which he "translated" into The Book of Mormon. {Early Mormonism and the Magic World View, Quinn}

April 6, 1830, the first meeting of the Church of Christ was called in New York; "Joe Smith" had his own church of "Mormonites". Smith also had a bit of a sordid reputation in New York. Eventually, circumstances led to Smith fleeing the state and heading for greener pastures in Kirtland, Ohio. {Rough Stone Rolling, Bushman}

Upon his arrival to Kirtland, Smith teamed up with an ex-Campbellite preacher named Sidney Rigdon. Rigdon was a preacher with multiple congregations and hundreds of parishioners at the time. There were roughly 5-7 families who were adherents to Mormonism in New York, but once Rigdon converted, many of his followers were also baptized into Mormonism, which provided a much-needed lifeline to keeping the young religion alive.

A few notable converts were Newell K. Whitney, Isaac Morley, and John Johnson, quite wealthy people with significant land and business holdings in the area. Smith quickly provided leadership roles for these men to fulfill their need to "serve the Lord," which manifested by them donating large sums of money, as well as various business ventures, to the church. {No Man Knows My History, Brodie}

Concomitant with the rising of Mormonism in Ohio was a concerted effort to settle "on the border with the Lamanites" (Missouri). Missouri was still the wild frontier west in the 1830s and seized land from Native Americans was cheap. Mormonism established a foothold there and eventually the Missouri congregations outnumbered that back at HQ in Kirtland.

Various factors led to Smith's excommunication from the Kirtland congregation. He fled in the middle of the night to escape vigilante justice by his previous parishioners and made it to Missouri.

Missouri in 1838, for Smith, was a place where he could finally put his theocratic motivations into practice. He formed an extra-military group, known as the Danites, to insure the security of the various Mormon settlements, numbering roughly 6-8,000 ppl at the time. After escalating conflict between the Mormons and Missourians, the Mormons raided and pillaged a few nearby settlements and razed them to the ground. Missouri militias responded by massacring the Mormons at a grain mill (Haun's Mill Massacre). Outnumbered and surrounded by multiple state militias, Smith and the Mormons surrendered. They were "exterminated" from the state of Missouri and found refuge in Illinois on the banks of the Mississippi. {The 1838 Mormon War in Missouri, LeSueur}

Nauvoo, Illinois became the new Mormon stronghold. Smith had learned from mistakes in Ohio and Missouri, resulting in his cultivating relationships with prominent politicians in Illinois. The city of Nauvoo was granted their own private militia which eventually outnumbered the militia of nearly any state in the Union at the time.

Brigham Young rose to prominence here. He, and the other apostles, went on a proselytizing mission to Europe in 1839, selling promises of American prosperity to any Englander willing to listen. Young established an immigration fund with a privately chartered ship, costing each European person a mere 4 pounds to immigrate to the Mormon settlement on the Mississippi. This resulted in thousands of converts making their way to the states. {Nauvoo, Kingdom on the Mississppi, Flanders}

Mormonism was a force to be reckoned with.

At the height of his career, Smith ran for POTUS in 1844 and was assassinated while being held in a jail. Nauvoo was roughly 12-20,000 Mormons (estimates are all over the place and real numbers are really hard to nail down) at this time, the population of Chicago, the next largest city in Illinois, was ~7-9,000 but growing fast.

Smith's death incited a schism crisis. A handful of men claimed to have the rightful authority to be the next one-true prophet. Smith's oldest son was 11 at the time so it wasn't reasonable for him to be the next prophet quite yet (see history of RLDS for further information). {Origins of Power, Quinn}

Some themes to tease out before an overview of Utah Mormonism: 1) persecution, 2) uniqueness, 3) galvanization.

1. Persecution works like fertilizer. Too little and the followers don't have a common enemy, too much and the movement is snuffed out (Catholicism pre-1550). But, just the right amount of fertilizer and the movement will grow and flourish. Mormonism hit that perfect sweet spot.

2. Uniqueness, Mormonism was wacky. Search newspaper databases for articles on Mormonism 1833-45 and many bear a title invoking "Deluded Fanatics," "Gold Bible Mormonites," or something of a similar derisive nature. Smith's claim as not just a preacher, but a prophet, a "mouthpiece of God," made Mormonism really stand out beyond most congregations.

3. Galvanization. Mormonism continued to make media headlines and Smith, along with other leaders, was able to turn that press into a persecution narrative, citing suffering in 1838 Missouri as their real-world examples of how real that persecution really was. A common group with a common enemy lives and dies together.

Utah Mormonism presents an interesting set of factors which caused Mormonism to become what it is today. Brigham Young rose to prominence during the schism crisis and took his group of 6-7,000 Mormons to an area where the U.S. Government would stop bothering them, Mexico. Utah Territory was carved out as a result of the Compromise of 1850, and the new Mormon theocracy had a place to flourish.

Life in Utah wasn't great for the Mormons in the beginning. It was an unsettled badland ruled by people with different beliefs, cultures, and skin color than them. Best estimates put Native American population in the territory around 20,000 when the Mormons got there.

While this issue is much more nuanced than a reddit post will justify, the Mormon/Native conflicts (1849-70s) resulted in near extinction of Native American people in Utah. It was only in 1980 when Native populations in Utah reached their pre-Mormon level of ~20,000 ppl. During this growing phase of Utah Mormonism, the European immigration fund remained successful. From 1847-1860, tens of thousands of European converts made the trek across the plains and settled with the growing Mormon settlement. Thousands of white religious fanatics flooding in while conflicts with Native raged and resulted in repeated decimation of the native tribes. These Europeans interbred and created their own self-contained economy with bartering and proprietary money; everybody relied on their neighbor to survive. {Indian Depredations, Gottfriedson}

Brigham Young was king of Utah. He was first Governor of the territory, director of the Office of Indian Affairs, and prophet of the Church; a trifecta of religious and government authority all held by one of the wealthiest Americans living at the time. {Blood of the Prophets, Bagley}

Polygamy also factors into Utah Mormonism more than we can imagine. It created a source of conflict with the U.S. Government that increased the impact of the 3 factors previously discussed. That, coupled with the geographic isolation of Utah, created a system where Mormonism couldn't help but flourish. It was the only authority in a land with dwindling Native populations and European religious fanatics that was well above the critical mass required to grow. {Origins of Power, Quinn}

Eventually, the United States Government disincorporated the Church and seized all assets >$50,000 until the practice of polygamy was officially renounced by the religion. It was {Official Declaration 1} and Utah was granted statehood in 1896. It only continued to grow from that point forward.

TL;DR Mormonism was started by a wacky and charismatic guy. Due to a number of complex factors, it had just the right algorithm to flourish into a major religion and build multiple theocracies. Once it attained the crucial "critical mass" of followers, nothing could stop it from continuing to grow.

Cred: independent researcher and podcaster of Mormon history.