Can someone help me to understand this difficult sentence structure?

In this example, the phrase placed between the em dashes is supposed to be read in parallel with the phrase "encouraged the transmission of." The sentence structure enables the author to compress two sentences into one, and to indicate that the second follows from the first. (I belatedly noticed that, in a comment beneath the OP's question, Jim says much the same thing I say here.)

In effect, you have this:

Gutman argues convincingly that the stability of the Black family encouraged the transmission of the Black heritage of folklore, music, and religious expression from one generation to another, a heritage that slaves were continually fashioning out of their African and American experiences.

and

Gutman argues convincingly that the stability of the Black family was crucial in sustaining the Black heritage of folklore, music, and religious expression from one generation to another, a heritage that slaves were continually fashioning out of their African and American experiences.

Because the stability of the Black family had the effect identified in the first sentence (namely, it "encouraged the transmission of the Black heritage of folklore, music, and religious expression from one generation to another"), it follows that the stability of the Black family also had the related effect identified in the second sentence (namely, it "was crucial in sustaining the Black heritage of folklore, music, and religious expression from one generation to another"). The original sentence that you quote says the same thing that my two sentences do, but in far less space.