From the "What We Believe" page of BLM's website (emphasis mine):

"We disrupt the Western-prescribed nuclear family structure requirement by supporting each other as extended families and “villages” that collectively care for one another, especially our children, to the degree that mothers, parents, and children are comfortable."

Surely you can see how someone might read the above and see how they would come away with thinking BLM is attacking the nuclear family, right? I don't necessarily agree with the position, but the Fox News segments with Tucker and Hannity write themselves based on BLM's own stated position.