What is the "tragic fact" to which Dr. King alludes?

The quote that you mention here comes from Martin Luther King, Jr.’s famous “I Have a Dream” speech.  It comes very close to the beginning.  In King’s own words, the “tragic fact” is that African Americans are not really free even 100 years after the end of slavery.  As King says,

But one hundred years later, we must face the tragic fact that the Negro is still not free.

King does not mean that blacks...

The quote that you mention here comes from Martin Luther King, Jr.’s famous “I Have a Dream” speech.  It comes very close to the beginning.  In King’s own words, the “tragic fact” is that African Americans are not really free even 100 years after the end of slavery.  As King says,

But one hundred years later, we must face the tragic fact that the Negro is still not free.

King does not mean that blacks are literally enslaved.  Instead, he means that they are figuratively enslaved by problems that face them.

Of course, King was well aware that slavery did not exist in the United States in 1963.  However, he said, the problems African Americans were so bad that they cumulatively made that race less than free.  King identifies three problems that still exist in the US.  First, he says, African Americans are harmed by racial injustice.  He says that they are

sadly crippled by the manacles of segregation and the chains of discrimination.

Second, he says that they are too poor when compared to other Americans.  He says that they live

on a lonely island of poverty in the midst of a vast ocean of material prosperity.

Finally, King says that African Americans are not fully accepted as true Americans.  I believe that this is very similar to the first problem that he mentions and that he includes this problem more for rhetorical purposes.  I think that he felt that three problems would sound better than two.  To him, this third problem was that African Americans were “still languishing in the corners of American society” and that they were still “exiles” in their own country.

Together, these problems make up the “tragic fact” that King referred to in his speech.  The “tragic fact” is that African Americans were not free even after slavery had been dead for a century.