When you have States with a high percentage of Blacks and have multiple White people running for a position and 1 Black. The fear or theory at the time was All the Blacks would vote for the Black candidate and the Whites would dilute their votes by spreading them among multiple candidates.
Runoffs arose to limit power of Black voters in Georgia
"At the Georgia State Capitol in the 1960s, State Rep. Denmark Groover led a charge for runoff elections.
A federal report looking at civil rights in America called Groover a "staunch segregationist" who pushed for runoffs in Georgia as a way to challenge a growing Black political power.
The report reads, "support for the majority-vote plan reinforced the moderate segregationist position. It did not remove anyone's right to cast a ballot, but it was commonly regarded as hampering African Americans—the stigmatized bloc voters.""
Racist lawmakers built Georgia’s election system, and now it’s affecting the balance of the Senate.
www.vox.com
"In 1963, state representative Denmark Groover from Macon introduced a proposal to apply majority-vote, runoff election rules to all local, state, and federal offices. A staunch segregationist, Groover’s hostility to black voting was reinforced by personal experience. Having served as a state representative in the early 1950s, Groover was defeated for election to the House in 1958. The Macon politico blamed his loss on “Negro bloc voting.” He carried the white vote, but his opponent triumphed by garnering black ballots by a five-to-one margin.
Groover soon devised a way to challenge growing black political strength. Elected to the House again in 1962, he led the fight to enact a majority vote, runoff rule for all county and state contests in both primary and general elections. Until 1963, plurality voting was widely used in Georgia county elections..."
"The method was popular across the former Confederacy: Alabama, Arkansas, Georgia, Mississippi, Oklahoma, South Carolina, and Texas all have primary election runoffs. As the Washington Post reported, just two non-Southern states have runoff rules, and those “almost never matter”:"