Great musical, great question. I was raised in the LDS Church (though am not currently a member), and have been actively studying church history (albeit as an amateur) for over 15 years, so can provide an overview.

Prior to 1978, the LDS Church officially prohibited black men from being ordained to the priesthood (in Mormonism, the lesser "Aaronic priesthood" is given to all worthy boys as young as 12, and the greater "Melchizedek priesthood" is given to all worthy men as young as 18), and prohibited both black men and women from performing temple ordinances (which are seen as crucial to salvation in Mormon doctrine).

Although Joseph Smith had previously ordained a black man (Elijah Abel) to the priesthood (and was planning to run for president on an anti-slavery platform before he was assassinated), the ban was implemented by Brigham Young in 1852, in a speech to the Utah Territorial legislature (early Utah politics were overtly theocratic), in which he said "any man holding any one drop of the seed of Cain...could not hold the priesthood." Early Mormonism in Utah is rife with these sort of statements - Brigham Young also believed that slavery was a divine institution, with both the "Mark of Cain" and "Curse of Ham" being invoked to explain why black people were destined for eternal servitude. The Book of Mormon contains a similar dynamic, in which the central villains of the narrative, the "Lamanites" (who were described as the ancestors of the American Indians), are also cursed with dark skin due to their unrighteousness.

This institutional racism overlapped with much of American society and didn't create much controversy for over a century. But once the civil rights movement kicked into high gear in the 1960s it began to attract a lot more notice. Some church members began advocating for change, church leaders seemed to split on the issue - there was a proposal to rescind the ban in 1969 by Hugh B. Brown (the least senior of the 3 top honchos) which was initially approved, then defeated once Harold B. Lee (honcho #2) returned. Shortly thereafter the church released a statement reiterating that the ban was of divine origin, but that at some point it would be lifted.

Throughout the 70s the furor only grew. Collegiate athletic teams often refused to play against BYU, and by the late 70s the IRS was also threatening to revoke the church's tax-exempt status had revoked Bob Jones U's tax-exempt status, and there were concerns that BYU could be next. This brings us to the 1978 invoked in the musical, when the head of the church received a "revelation" from God and lifted the ban. Current Mormon policy is non-discriminatory when it comes to race.

This topic remains a fraught one in LDS culture today, which I can expand on a bit if it doesn't run afoul of the 20-year rule.

Sources:

Mauss, Armand L. (2003). All Abraham's Children: Changing Mormon Conceptions of Race and Lineage.

Journal of Discourses

lds.org essay "Race and the Priesthood"

Edited to clarify current policy and get the correct century.

Initially the LDS church was fairly progressive toward African Americans. It allowed black members. It even had some black men in positions of moderate leadership. For example Walker Lewis was an important Elder in the Boston branch of the church. When he ran for US president in 1844, church founder Jospeh Smith, even had a campaign promise to emancipate all the slaves, and compensate their owners via the sale of public lands.

That changed when Brigham Young gained the church leadership. I'm going to extensively quote a speech he gave to open the Utah legislative session on Feb 5th 1852 highlighting the church's positions on blacks.

Young taught that Africans were the descendants of the original murderer Cain.

What is that mark? you will see it on the countenance of every African you ever did see upon the face of the earth, or ever will see. ... I tell you, this people that are commonly called negroes are the children of old Cain.

He also taught that this meant they were cursed and not allowed to hold priesthood or fully participate in church.

Now then in the kingdom of God on the earth, a man who has has the Affrican blood in him cannot hold one jot nor tittle of preisthood; Why? because they are the true eternal principals the Lord Almighty has ordained, and who can help it, men cannot. the angels cannot, and all the powers of earth and hell cannot take it off, but thus saith the Eternal I am, what I am, I take it off at my pleasure, and not one partical of power can that posterity of Cain have, until the time comes the says he will have it taken away. That time will come when they will have the privilege of all we have the privelege of and more. In the kingdom of God on the earth the Affricans cannot hold one partical of power in Government. The the subjects, the rightfull servants of the resedue of the children of Adam, and the resedue of the children through the benign influence of the Spirit of the Lord have the privilege of seeing to the posterity of Cain; inasmuch as it is the Lords will they should receive the spirit of God by Baptisam; and that is the end of their privilege; and there is not power on earth to give them any more power.

They could join and be baptized into the church but that was the limit. They couldn't teach, administer, or participate in any other way other than simple membership.

This curse of being black was so great that he said it'd be a favor to kill a man who was "mingling seed" with Africans

Where the children of God to mingle there seed with the seed of Cain it would not only bring the curse of being deprived of the power of the preisthood upon themselves but they entail it upon their children after them, and they cannot get rid of it. If a man in an ungaurded moment should commit such a transgression, if he would walk up and say cut off my head, and kill man woman and child it would do a great deal towards atoneing for the sin. Would this be to curse them? no it would be a blessing to them. -- it would do them good that they might be saved with their Bren. A man would shuder should they here us take about killing folk, but it is one of the greatest blessings to some to kill them.

He then goes on to say that they should be barred from government.

Again to the subject before us; as to The men bearing rule; not one of the children of old Cain, have one partical of right to bear Rule in Government affairs from first to last, they have no buisness there. this privilege was taken from them by there own transgression.

The are also natural slaves, according to God's orders, but they shouldn't be treated like they were under southern chattel slavery.

I am as much oposed to the principle of slavery as any man in the present acceptation or usage of the term, it is abused. I am opposed to abuseing that which God has decreed, to take, a blessing, and make a curse of it. It is a great blessing to the seed of Adam to have the seed of Cain for servants, but those they serve should use them with all the heart and feeling, as they would use their own children, and their compassion should reach over them, and round about them, and treat them as kindly, and with that humane feeling necessary to be shown to mortall beings of the human species.

This speech was far from the only time Young discussed the issue. But he really spelled it all out on that occasion. I could have pulled from dozens of other speeches or talks he gave. His 26 volume Journal of Discourses are full of such talks.

Blacks were sinners and descendants of the original sinner Cain. They were cursed to be servants, and they had no proper role in the governance of the church or civil affairs. As Young lead the church for 30 years, these doctrines became firmly entrenched within the church.

For example here is an except from a talk given by Young's successor John Taylor.

And after the flood we are told that the curse that had been pronounced upon Cain was continued through Ham's wife, as he had married a wife of that seed. And why did it pass through the flood? Because it was necessary that the devil should have a representation upon the earth as well as God; and that man should be a free agent to act for himself, and that all men might have the opportunity of receiving or rejecting the truth, and be governed by it or not according to their wishes and abide the result;

Similar talks were given over the following decades. Going further forward to the 1950s and 1960s we have church leaders like John Lund publishing books like The Church and the Negro or Bruce McConkie's Mormon Doctrine spelling out how this was still official doctrine and belief.

The problem for the church was that of course by the 1950s the Civil Rights movement was starting to take hold. The LDS church was under increasing scrutiny and pressure. You still had church leaders like Mark Peterson giving talks that included snippets like this.

Think of the Negro, cursed as to the Priesthood.... This Negro, who, in the pre-existence lived the type of life which justified the Lord in sending him to the earth in the lineage of Cain with a black skin, and possibly being born in darkest Africa – if that Negro is willing when he hears the gospel to accept it, he may have many of the blessings of the gospel. In spite of all he did in the pre-existent life, the Lord is willing.... to give him the blessings of baptism and the gift of the Holy Ghost. If that Negro is faithful all his days, he can and will enter the Celestial Kingdom. He will go there as a servant, but he will get celestial glory.

The LDS church was still actively teaching that African Americans were cursed, and that heaven was segregated. In the 1950s the LDS church was far from alone in discussing race in such terms. But by the late 1960s and early 1970s they were more of an anomaly and the civil rights leadership increasingly put them in the spotlight. The church owned school BYU became the target of protests and boycotts. Some schools like Wyoming responded by expelling athletes who wouldn't compete against BYU. Others like Stanford joined the protests and officially refused to play the school. At the same time the IRS started cracking down on various church owned affiliates for engaging in racial segregation, starting with the non-LDS Bob Jones University.

The problem the LDS Church had was that they had over a century of prophets saying that the LDS Church's policies and doctrines came directly from God via revelation. It wasn't simply a policy that could be changed... God had to change it. From the writings of the church leadership in the 1970s you can see how much many of them wanted to change it. In 1969 the Quorum of Apostles even voted to change the policy. Then the Apostle Harold Lee who had been traveling in Europe returned to Salt Lake and reminded them that they couldn't just vote it out... so it stayed.

In 1973 Spencer Kimball became the LDS prophet. He had been among the group who had voted to change the policy in 1969, and was considered a liberal. He began laying the groundwork for a policy change. First he instructed church historian Leonard Arrington to investigate the history of the policy. Arrington was able to show that it hadn't been in place since the beginning as had been commonly thought, and instead that it was only implemented after founder Jospeh Smith's death. He documented that at one point certain black members had held the priesthood. Kimball then instructed Apostle Bruce McConkie to research all of the revelations and doctrinal statements made by prior church leaders. McConkie was able to find several talks that indicated that "someday" the curse might be lifted by God.

Armed with history and doctrinal backing Kimball went to the Quorum of Apostles in 1978 and informed them that he intended ask God to reverse the policy. Weeks later he returned and informed them that he had received revelation that God had heard his prayer and revealed a new policy to him. Having already laid the groundwork, Kimball was able to convince the Quorum to join him and remove segregation in the church. A new policy was announced in which all male members of the church could receive the priesthood regardless of race, and all worthy members regardless of race could attend the temple and participate in all the rituals of the church.

For further reading I would suggest.

• W Paul Reece Religion of a Different Color: Race and the Mormon Struggle for Whiteness
• Armand Mauss All Abraham's Children: Changing Mormon Conceptions of Race and Lineage
• Lester Bush, Armand Mauss Neither White Nor Black: Mormon Scholars Confront the Race Issue in a Universal Church

It's worth noting that this "echo" of Brigham Young et al's teachings still echo in teachings today.

For example, here's what Young said about interracial marriage (source):

"Shall I tell you the law of God in regard to the African race? If the white man who belongs to the chosen seed mixes his blood with the seed of Cain, the penalty, under the law of God, is death on the spot. This will always be so.”

This is what the church said on the cusp of the civil rights movement (source):

"God has commanded Israel not to intermarry. To go against this commandment of God would be in sin. Those who willfully sin with their eyes open to this wrong will not be surprised to find that they will be separated from the presence of God in the world to come. This is spiritual death… To intermarry with a Negro is to forfeit a Nation of Priesthood holders."

And here's what LDS.org says today:

“We recommend that people marry those who are of the same racial background generally, and of somewhat the same economic and social and educational background (some of those are not an absolute necessity, but preferred), and above all, the same religious background, without question."