OK.  The slogan "black lives matter" is used to remind non-black people in the US that black lives matter just as much as any other lives.  Historically, in our country, we have chosen to devalue the lives of black people.  Responding to that slogan by saying that "all lives matter" (which is true, and what the BLM movement is trying to underscore) is, in essence denying the premise that the lives of black people are just as precious as the lives of anyone else.  
I don't understand why some find that slogan so offensive.  Why would the response not be: "of course, they do, and we will do everything that we can to ensure that the lives of black people are accorded the same value as the lives of all others."?