not a teacher
I'm assuming you understand the phrase "walk the walk" and how it relates to "talk the talk". For example: "When it comes to civil rights, a lot of people just talk the talk but don't walk the walk."
In your quote it means that they not only talked about civil rights and what should be done, they also went out and tried to do something about it, as Dr King did. In this case it could also refer to literally walking with him on the famous protest marches.
So yes, "They, together with Dr King, walked the walk."
Interested in Language