I wonder about this: there are a great many works that influence people of a certain time and place such that they almost become givens of cultural consciousness. For example, "to be, or not to be?" from Hamlet and "I know why the caged bird sings" from Paul Lawrence Dunbar's Sympathy and more so from Maya Angelou's autobiography and also her poem, which share the title, I Know Why The Caged Bird Sings.
The second example gets more to the meat of what I've been thinking about. English is influenced by a robust popular culture and consciousness that relies heavily on referencing other works as a way of conveying ideas and themes in a variety of arenas. English is not at all unique in this respect, but because of its global reach, the sheer amount of works written in or conceived of in English and translated into the language, I feel that English is influenced by the power of cultural references in ways that other languages may not be.
I'm not a literary scholar and to be honest, a lot of subject exclusive scholarly work kind of turns me off, but when literary scholarship interacts with social phenomena, that starts to turn wheels for me. After all, literature is not really "for" scholars, but for lay people-- something that I think critics, scholars and other academics tend to forget.
Anyway, I don't think you're alone in your exposure to MLK's speech. In fact, I would be willing to posit that a lot of people are in the same boat. I just find it interesting that these "great works" seem to cause these ripples of awareness, which seem to form the backdrop of the way that people tend to interact with each other by influencing assumptions of what people "should" be familiar with and to what degree.