In order to answer this question, you simply need to look at the words of Dr. King’s speech. He clearly lays out what he sees as evidence that African Americans were not free. Here is the relevant part of the paragraph:
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.
So what does this all mean?
First, he is saying that there is still segregation. This was true in 1963 as it was still legal to keep blacks segregated in restaurants, movie theaters, and other such things. Second, he is saying that there is discrimination. Discrimination occurs when we treat people differently based on something like their race. This would have occurred, for example, if bosses refused to hire blacks for certain jobs. This was clearly still happening in 1963. Third, he is saying that America as a whole was a rich country while African Americans remained poor. This was true in 1963 and is still true (though less so) today. African Americans were and are more likely to be poor than whites are. America is rich and many African Americans do not share in that wealth. King’s final claim is that blacks “languish in the corners” of our society and are “exiles” in their own country. This is a little harder to discuss because he is not making a specific claim. He is simply saying that African Americans are not allowed to be part of the mainstream of American society. This is probably true, but is not really as measurable as things like black poverty rates.
So, the evidence that King is presenting consists of these four claims. He asserts that there is still segregation and discrimination. He says that blacks remain poor as the country is rich, and he says that blacks are relegated to a lower status in our society.