from your link:
Howard W. Smith, a Democrat and avid segregationist from Virginia, indicated his intention to keep the bill bottled up indefinitely.
Johnson, who wanted the bill passed as soon as possible, ensured that the bill would be quickly considered by the Senate. Normally, the bill would have been referred to the Senate Judiciary Committee, chaired by Senator James O. Eastland, Democrat from Mississippi. Given Eastland's firm opposition, it seemed impossible that the bill would reach the Senate floor. Senate Majority Leader Mike Mansfield (a republican) took a novel approach to prevent the bill from being relegated to Judiciary Committee limbo.
The bill came before the full Senate for debate on March 30, 1964 and the
"Southern Bloc" of 18 southern Democratic Senators and one Republican Senator led by Richard Russell (D-GA) launched a filibuster to prevent its passage.
On the morning of June 10, 1964, Senator Robert Byrd (D-W.Va.) completed a filibustering address that he had begun 14 hours and 13 minutes earlier opposing the legislation.
further reading
On August 29, 1957, President Dwight Eisenhower signed the Civil Rights Act of 1957 into law. One of the main goals of the act was to help ensure that all African-Americans were ensured their right to vote. It set up the Civil Rights Commission to monitor civil rights abuses. Unfortunately,
the bill lost a lot of its teeth due to opposition from southern Democrats. The resulting bill was not universally lauded as many such as Ralph Bunche felt it was worthless in achieving its aims. However, it became the first of many bills and measures to help fight against Jim Crow laws and anti-voting actions in the South.