How did the voting rights act of 1965 stop discrimination in areas where voter eligibility tests were previously used?

It required federal supervision.  Many Africans were prevented from voting because they were being harassed.  To ensure that they too are able to vote, the Federal government instructed law enforcement and the National Guard to help African Americans to register and vote without fear of being driven of the voting stations.
 

Of course it could have been avoided. a great many scholars argue it happened because of truman's use of fear to gain support. his predecessor fdr was the most popular and charismatic president ever. truman was a machine politician chosen as the least offensive choice around for vp. had henry wallace remained vp and then president, the cold war might not have happened. or had dewey been defeated in the primaries by robert taft, and then taft defeated truman in 48, the cold war also might not have happened. taft was somewhat isolationist, opposing nato and the korean war. wallace believed truman to be unnecessarily provocative in his dealings with the soviets.of course for strong anti communists, they claim the soviets would see any sign of weakness and try to take over the world. but this is a conspiracy theory not backed by evidence. most communist movements were local and not controlled by moscow. stalin had just lost over 20 million people and was exhausted and paranoid. his number one fear was a rearmed germany allied with the us, and truman made his nightmare real with the marshall plan. truman's blunders are at least half responsible for the cold war and neither wallace nor taft would have made them.on top of that, stalin was dying and in poor health. his successor khrushchev was a reformer who ended the forced labor camps, set many political prisoners free, criticized stalin, and declined to build a powerful soviet navy to confront the us. stalin actually proposed a neutral united germany. a paranoid american gov't turned him down, guaranteeing a continued cold war for decades more.