Wikipedia tells me that the Dictator of Zaire volunteered to host it. But why would the almost entirely American event agree to that? Surely there were more convenient places that would command higher ticket prices available. What led to those involved accepting this offer?
Short answer: he had the money and it was customary for matches to take place all over the world.
Long answer: It's 1967 and Muhammad Ali, already established as one of the best boxers of all time and reigning heavyweight champion, is banned from professional boxing and stripped of his title due to his refusal to be drafted in the Vietnam War. But as time goes on and public opinion shifts against the war in Vietnam, the public grows more sympathetic to Ali, and he returns to the ring in 1970. Unfortunately, in his years out of the ring, Ali has aged and grown slower, and he loses the Fight of the Century in 1971 to then-heavyweight champion Joe Frazier. The hero has lost his lustre. He is only 29.
1973. George Foreman, aloof, antisocial, and newly 23 years old, becomes heavyweight champion after defeating Joe Frazier in Kingston, Jamaica in a technical knockout. In Foreman's first defence of his new title in Tokyo, he takes out his opponent in 2 minutes, one of the fastest knockouts in Heavyweight Championship history.
1974. Frazier and Ali rematch in a non-title match in Madison Square Garden, having both lost their titles - Ali to Frazier, and Frazier to Foreman. But this time, something is different. Although now 32, Ali has adapted his strategy - he uses clinching extensively, a defensive move boxers use when they are tired or need a break from an onslaught - essentially wrapping your arms around theirs to hold them up and force a pause. Although Ali reportedly clinches 133 times, he wins.
This brings us to the question of Ali vs Foreman. Foreman, a notoriously heavy hitter, is 25 and reigning heavyweight champion. Ali, his senior by 7 years, hasn't held the title since 1967, but his defeat of Frazier and enduring popularity create massive public interest in an Ali vs Foreman fight. Knowing this interest, both men demand $5M in payment, which would be the largest purse in history for a fight.
Many rival promoters try to requisition the $10M necessary to pay the fighters, but to no avail, until finally legendary promoter Don King signs contracts with each of them for the amount requested. The fight is on: the Rumble in the Jungle. But there's one snag: he doesn't have $10M either.
So begins a process of seeking sponsorship. It just so happened that Zaire's president, Joseph Mobutu, was looking for something to sponsor at the same time: his regime had overthrown and assassinated Patrice Lumumba, an anti-colonial leader, enraging American civil rights leaders who had looked up to Lumumba, and he sought to win their favour back. Sponsoring a major event was just such an opportunity, and the $10M needed was a paltry sum considering the massive western investment in resource extraction in his country. (Mobutu had a habit of looting state funds to deal with issues.) In fact to further promote a cultural exchange, Mobutu also agreed to host a 3-day music festival leading up to the fight, featuring African and African-American musicians and organized by famed South African trumpeter Hugh Masekala.
In later years, it would be pointed out the irony of Ali, a member of the Nation of Islam and black nationalist, participating in an event paid for by a corrupt dictator trying to rebuild his reputation after murdering a respected African nationalist leader. But in 1974, all that seemed to matter to Ali was the crown. Odds against him but with the crowd firmly on his side, it was clear Ali had refined his defensive technique, and it was in this fight that he introduced his now-famous rope-a-dope technique - lying against the ropes, allowing your opponent to throw punches where most of their strength is absorbed by the elasticity of the rope, tiring them out. Midway through the fight after allowing Foreman to pummel him wildly, Ali brought his energy level up against the now-exhausted Foreman, who would later recall: "I thought Ali was just one more knockout victim until, about the seventh round, I hit him hard to the jaw and he held me and whispered in my ear: 'That all you got, George?' I realized that this ain't what I thought it was." Ali won by knockout in the 8th round.
The Rumble in the Jungle was the most watched television event in history and brought in an estimated $100M in global revenue, and it probably wouldn't have happened if it weren't for a corrupt African dictator.
##References
I want to push back on an assumption implicit in this question. The Democratic Republic of Congo has a reputation in the 21st century for being a very violent and poverty-stricken country.
However, don't assume that such a situation can be traced back to the beginning of time.
In fact, in 1974 when the match took place, Zaire was at the end of a decade-long period of robust economic growth on the back of mining revenues. To give a very rough sense of where Zaire stood in 1974, lets look at GDP per capita for Zaire and select other countries (Zaire appears as Democratic Republic of Congo on this table). As you can see, Zaire's GDP per capita compared very favorably to countries like Cambodia, Laos, India, China, Pakistan or Mongola. As of 1974, Zaire's GDP per capita was similar to that of Honduras or Kenya.
And in terms of violence, Zaire had faced major outbreak of violence in the Katanga secession crisis from 1960-63 and during the Simba revolt in 1964-65. However, there was not any sort of civil war or inter-state war until the Shaba invasion in 1977.
The reputation for widespread violence that the DRC now has is traceable to decay of the state and armed forces in the 1980s and a decade of crisis where the 1994 genocide in Rwanda set up conditions for a violent civil war in Zaire in 1997 that overthrew Mobutu, and then a second war that raged for 6 years and may have killed up to 5 million people.
I do not wish to present a pollyana picture of Zaire in 1974. There certainly was poverty in the country, as well as political repression by Mobutu's regime. But, it would be anachronistic to describe Zaire in 1974 as "one of the most violent, poverty stricken countries on Earth".
Further Reading
The Rise and Decline of the Zairian State by Crawford Young and Thomas Turner. 1985, University of Wisconsin press.
Historical Dictionary of the Democratic Republic of the Congo, Third Edition by Ezimet Francois Kisangani and F. Scott Bobb (eds). 2010, Scarecrow Press. "Economy"
the Congo; Plunder and Resistance by Renton, Seddon and Zeilig. 2007, Zed books.