In this excerpt from the "Letter from Birmingham City Jail," Martin Luther King Jr. is criticizing the white Southern clergymen who had criticized the civil rights movement for taking direct action against racial discrimination and injustice.
In an open letter, they had argued that a purely legal strategy was the only one to adopt to achieve civil rights. That meant there was no place for the kind of large-scale public demonstrations, bus boycotts, lunch-counter sit-ins, and other forms of direct action that the civil rights movement under King had adopted.
Although the white clergymen had claimed to support the aims of the civil rights movement, King describes their acceptance of those aims as "lukewarm", meaning half-hearted. The clergymen may claim to support the civil rights movement, but by refusing to endorse direct action their support is actually damaging to the cause.
Outright rejection of the civil rights movement—the kind displayed white supremacist politicians and organizations such as the Ku Klux Klan—is something that King and his followers can deal with as it makes perfect sense, even if it is morally reprehensible. But the "lukewarm acceptance" by the white clergymen of the civil rights cause is much more bewildering because it makes no kind of sense at all. They claim to be in favor of civil rights and yet are not prepared to do whatever it takes to make them a reality, despite the fact that they're Christians and should therefore understand the necessity of challenging injustice, even if it means breaking the law.