I've seen/heard that this speech was all but irrelevant for the rest of his lifetime and the years after his death. Why do I and everybody I know know of the speech today?
edt: source is drew hansen's book "The Dream: Martin Luther King, Jr., and the Speech that Inspired a Nation", but as used by another secondary analysis (http://www.theguardian.com/world/2013/aug/09/martin-luther-king-dream-speech-history)
the article is super interesting and talks in depth about MLK/his advisor's thoughts going into the march but not of how the speech became so well known today.
The speech actually quickly became very famous, since he was already one of the best known civil rights figures and was well known for his speaking ability. It was well known then for the same reasons it was well known now - it's one of the most moving, stirring speeches in American history.
source: Parting the Waters, Taylor Branch