The italicized part is a nonrestrictive (whiz-reduced) relative clause modifying the sentence subject. It plays the syntactic role of adjunct. It's the same thing as writing these:
Eager to be true partners in a struggle for social justice, African Americans throughout the South responded with great hope and enthusiasm.
African Americans, eager to be true partners in a struggle for social justice throughout the South, responded with great hope and enthusiasm.
African Americans throughout the South, who were eager to be true partners in a struggle for social justice, responded with great hope and enthusiasm.
Once you delete so much of the clause structure it can float more freely in the sentence, in the same way that adverbials can. But it is nonetheless modifying the noun phrase in the subject at the front, even when it falls at the end as in your version.
There's a lot of confusing terminology possible with these, such as absolute adjectivals, adpositive constructions, discontinuous constituents, and predicative adjuncts. I don't know that those help people more than confuse them, though, especially the latter because it is meant to apply to the predicand not the predicate proper.