Think of it as someone who is supposed to staying alert for guard duty or while driving a car. If they fall asleep, they will get a rude, meaning rough, awakening if they are awakened by the enemy attacking or by their car crashing. So King is saying that if the U.S. thinks that the civil rights movement is going to go away and they stop paying attention to injustice and racism, they will have a rude awakening in the form of continued revolt.