WWII was a catalyst for the civil rights movement

• the fight against fascism sharpened the contradiction between segregation and the American claim to be one of the great democracies; as a result of this Truman issued an executive order prohibiting racial discrimination within the army, before that Roosevelt established the Committee on Fair Employment Practices to investigate complaints on discrimination in he defense industries.

• the issue really became important once the cold war tensions cooled down a bit (relative to the early fifties) and Lyndon Johnson decided to break with the south and adopt civil rights as part of his agenda.

It's probably important to distinguish between two phases of the civil rights movement, at least legally. There's the judicial phase, and the legislative phase.

The judicial phase ends with Brown v. Board of Education, 347 U.S. 483 (1954), the landmark Supreme Court decision that ended school segregation -- at least in theory. That decision came in 1954, rather than ten years earlier or ten years later, because it was the culmination of a concerted legal effort led by Thurgood Marshall that forced the Court to confront Plessy v. Ferguson's "separate but equal" theory, and ultimately stretched that theory to its breaking point. I apologize for the Wikipedia link, but outside of a constitutional case book, this page is actually a great summary of Marshall's legal career in the years leading to Brown. Some highlights:

• Sipuel v. Board of Regents of University of Oklahoma, 332 U.S. 631 (1948): state with a white-only law school must provide for equal education of black students.
• Sweatt v. Painter, 339 U.S. 629 (1950): separate graduate schools were not equal, because of the networking value of a graduate education, and the state's separate blacks-only graduate program violated the equal protection clause.
• McLaurin v. Oklahoma State Regents, 339 U.S. 637 (1950): relegating a black student, admitted to the University of Oklahoma under a previous court decision, to deliberately inferior on-campus facilities violated the equal protection clause.

In this manner, Brown built on all that came before it, which was necessary legal prologue.

The legislative phase begins with Brown and ends with 1965's Voting Rights Act. Between 1954 and 1965, the South did everything within its power to oppose desegregation, aided by the fact that the Supreme Court is ill-suited to enforce its own orders, requiring clarification in later opinions related to enforcement. See Brown v. Board of Education of Topeka (Brown II), 349 U.S. 294 (1955) (directing that desegregation take place "with all deliberate speed"). By 1955, then, the Supreme Court has outlined:

1. That segregation is illegal,
2. That it should be remedied,
3. And how it should be remedied on a case-by-case basis, with district courts taking continuing jurisdiction of segregated schools and overseeing necessary remedies, like busing, etc.

Necessarily, though, this is still a slow process. One of my favorite books on the subject -- about the inertia in translating judicial desegregation into results through legislative action -- is Robert Caro's Master of the Senate, which chronicles then-Senator Lyndon Johnson's ascent to power and relationship with the "Solid South," which stood against any legislative desegregation effort. Caro would tell you that even presented with a Supreme Court mandate, the Senate still slowed down any real civil rights reform by insisting that states should be able to fix themselves. I don't think this is a full answer, but it does help explain why real civil rights victories came slowly even after Brown.

The premise of your question isn't necessarily sound. The Civil Rights Movement may have achieved its greatest recent victory with the 1965 Civil Rights Act, but one suspects that the 'civil rights movement' - as in, African Americans striving for equal civil, economic, social, and cultural rights - has been a virtual constant since John Hawkins brought over his first 500 slaves in 1565.