Anaphora is, according to eNotes, "the repetition of a phrase at the beginning of sentences," a technique seen throughout King's famous speech. Anaphora is seen in many iconic speeches--it's a stylistic move that brings emphasis to the repeated phrase. Let's take a look at the paragraph you've indicated (full text available at AmericanRhetoric.com):
But one hundred years later, the Negro still is not free. One hundred years later, the life of the Negro is still sadly crippled by
the manacles of segregation and the chains of discrimination. One hundred years later, the Negro lives on a lonely island of poverty in the midst of a vast ocean of material prosperity. One hundred years later, the Negro is still languished in the corners of American society and finds himself an exile in his own land.
In this anaphoric paragraph, Dr. King brings up a series of injustices being faced by the African American people in their own country. The impact of this list is already significant; the listener is forced to face and consider the realities of the way American society is suppressing and failing a large portion of its population.
The true rhetorical power, though, comes from the anaphora, "one hundred years later," which prefaces each item on the list. It gives context to these horrors--not only have these unfair, horrid things been happening, but they've been happening for a century. In particular, they are happening a century after African Americans were proclaimed "free"! The effect of this emphasis is an overwhelming sense of how vast and persistant injustice has been in the United States. King is not bemoaning some passing fad, but instead declaring that there's a whole century's worth of national sins that have been committed against his people. The anaphora drives the point home that this injustice has been a part of American history and society for the entire century prior to the speech, and intensifies the feeling that King's call to action will, and must, change this history.