Actually I think the logic of BLM is quite obvious and compelling.  If you were to fill in what is implied in the shorter slogan, it would go something like this: All lives should matter, but historically in the United State Black Lives have mattered less (See slavery, Jim Crow, etc.), therefore we need to proclaim that Black Lives Do Matter also.  BLM does not at all claim that one race has privilege over another, but that all races should have the same privileges and rights that currently not all do.