The quote is as follows: "We must accept finite disappointment but never lose infinite hope." It means that, in the short term, we may not achieve our goals, but, if those goals are noble, we will succeed. Therefore, we should never lose sight of them.
In the context of the Civil Rights Movement, King meant that temporary disappointments should not deter people from the pursuit of civil rights. The 1960s were particularly trying in this effort. King delivered this quote during a speech on February 6, 1968, nearly two months before his assassination in Memphis, Tennessee on April 4, 1968. I do not think that it would be wrong to state that he would have been disappointed with the aftermath of his assassination, which included riots in numerous cities throughout the country. He would have likely deemed this an unwillingness to "accept finite disappointment." Others referred to the events, sometimes called "Holy Week Uprising," as a revolt against consistent and unwavering systemic oppression.
There were numerous disappointments before King's assassination. The Civil Rights Act of 1964 was slow to pass and required frequent pressure on President Johnson, who stalled to avoid upsetting long-standing incumbents in the Southern Coalition.
The Voting Rights Act of 1965 was passed, but there had already been numerous efforts to obstruct black voting power. At the Democratic National Convention in 1964, Fannie Lou Hamer, a leader within the Mississippi Freedom Democratic Party, tried to unseat the official Mississippi delegation. Her desire was to address black disenfranchisement and to demand parity within the party. She also used the platform of the convention to try to talk about what it was like for black people to attempt to register to vote or go to polling places in the Deep South. Her speech, which was televised, was interrupted by President Johnson, who aired an "impromptu press conference" during the speech. He very likely did not want voters to know about the many ways in which the South prevented black people from voting, including citizenship tests. To address this injustice, Dr. King, his wife Coretta Scott, future Congressman John Lewis, and many others marched into Selma, Alabama in 1965.