First we must delve into two topics:
- Joseph Smith's candidacy for POTUS
- Missouri Executive Order 44
It's very possible that Brigham Young disallowed those of African descent to receive the priesthood due to the persecutions from Missouri. The abolition of slavery was Joseph Smith, Jr.'s, platform for presidency, and that is what ultimately led to the formation of the mob at Carthage. In the Encyclopedia of Antislavery and Abolition by Peter P. Hinks and John R. McKivigan, an excerpt from one of Joseph's speeches is given:
Petition, also, ye goodly inhabitants of the slave states, your legislators to abolish slavery by the year 1850, or now, and save the abolitionists from reproach and ruin, infamy, and shame.
Pray Congress to pay every man a reasonable price for his slaves out of the surplus revenue arising from the sale of public lands, and from the deduction of pay from the members of Congress.
Break off the shackles from the poor black man, and hire him to labor like other human beings: "for an hour of virtuous liberty it is worth a whole eternity of bondage" ...
Restore freedom! break down slavery! Banish imprisonment for debt, and be in love, fellowship, and peace, with all the world!... and when we have the red man's consent, let the Union spread from the east to the west sea. ...The Southern people ... will help to rid so free a country of every vestige of slavery, whenever they are assured of an equivalent for their property.... Were I the president ... when that people petitioned to abolish slavery in the slave States, I would use all honorable means to have their prayers granted, and give liberty to the captive by paying the Southern gentleman a reasonable equivalent for his property, that the whole nation might be free indeed!
By the time that the members of the Church left Missouri, the general state population felt that they had every right to go after them; the abolition of slavery would mean that whites would have to work for themselves (Heaven forbid!), and there was the rumor (courtesy of excommunicated adulterer John C Bennett) that Porter Rockwell (Joseph Smith's bodyguard) had tried to assassinate Missouri Gov. Lilburn Boggs. This (along with the Battle of Crooked River) led to Missouri Executive Order 44 (aka Mormon Extermination Order), issued on 27 October, 1838. According to this executive order (which was unconstitutional, by the way), every member of the Church was considered an enemy to the state, and therefore could be legally killed on sight, no questions asked.
EO44 remained legally on the books in Missouri until 25 June, 1976. The Priesthood Ban on those with African heritage was lifted on 30 September, 1978. Missouri, one of the more racist US states towards African-Americans (even to this day), could have easily made martyrs of any local African-American that had received the Priesthood (and thereby have the potential to receive stewardship over white members as bishops, stake leaders, seventies, and apostles [since LDS congregations were not segregated]) without legal repercussion until the rescission of EO44.
Some may ask, "Why not just put a ban on African-Americans who only live in Missouri?" Mainly because this does not solve anything. Once one receives the priesthood, it cannot be rescinded except by committing a grave sin. So, then, what would happen to those African-Americans that received the priesthood and then (for some reason) decided to move to Missouri? Many people move to many places, and there have surely been some black LDS families who have moved to Missouri since the time that EO44 was written.
I personally think that it was wise of Heavenly Father to allow the Priesthood Ban to be in effect for as long as He did. By doing so, He protected every African-American Church member (as well as every other Church member by affiliation) that had moved to or converted in Missouri from potential violence (which would have been completely legal under Missouri law).
Is this the only reason why the Ban was placed? I don't know. I'm not even sure that it crossed the minds of the prophets of the time; they may not have even been given any reason at all. There are many instances in the scriptures where prophets are commanded to do something without knowing the real reason why. Sometimes the answer is revealed to them in this mortal life, and sometimes they may have to wait until the next.
It may even be that this was purely a human error, considering that we do not consider prophets to be omniscient or infallible.
Unfortunately, Brigham Young personally opined that it was because of their supposed ancestory (descendants of Cain) that African-Americans could not hold the Priesthood. Many members mistook his opinions for doctrine, leading them to adopt such opinions themselves during that time. The adoption of the "descendants of Cain" as an excuse was a flaw of the Church Presidency of the time, though it may have also helped to stave off persecution related to the abolition of slavery. Luckily, the views of most LDS members today are much more open-minded, and are not as prone to confuse the private opinion of a prophet with the official doctrine that he preaches.
Why would a black person want to join the Church of Jesus Christ of Latter-day Saints?
Because it is an inspired faith that cares about its people equally, and while some members may make a huge sacrifice, it may be a sacrifice that helps themselves and everybody else in the big picture. It is the one true church that Christ Himself established upon the earth, the same as He has done in the past through various other prophets.
I hope that this all helps!
Hinks, Peter P., and John R. McKivigan. “Church of Jesus Christ of Latter-Day Saints and Antislavery.” Encyclopedia of Antislavery and Abolition, 1, Greenwood Publishing Group, 2007, pp. 159–160.