We are caught in an inescapable network of mutuality, tied in a single garment of destiny. king includes this sentence to draw attention to the interrelationship of events. illustrate the connectivity of his professional network. challenge the concept of predetermined destiny. explain his plan for a peaceful demonstration.

For the answer to the question above, the quote is that we are caught in an inescapable network of mutuality, tied in a single garment of destiny." I think he means that our lives are unavoidably connected with each other -- like a network of threads that can't really be untied/unknotted, and that we share a common destiny, whatever it may be. As much as some people may want to believe and live their lives as if they are completely alone and independent of other people -- or as part of one group that can remain isolated from other groups -- he is pointing out that our fates are necessarily intertwined to some extent. He seems to be using a fabric metaphor-- a network of threads becomes a single garment which represents our collective destiny as a society. 
You can kind of see an example of this in the economic situation in the world today. It is hard to find a place in the world today where individuals are not experiencing some effect of this economy, which is tied to gas prices, which is tied to food and commodity prices, which is tied to people paying their bills, which is tied to foreclosures, which is tied to big banks asset portfolio, which is tied to credit availability, which is tied to investor confidence, etc Each of us is affected somewhere along the line.