Essentially, the entire movement is based around the fact that blacks, more than any other skin colour, are STILL persecuted in the US, above what is necessary, by police.
The whole, "well, all lives matter," and, "if you don't care about everyone, then you're giving importance to black people," arguments, are trying to include others in said movement, when really emphasis must be placed on how blacks are treated.
Absolutely, nobody is more important than anyone else, due to skin colour, however white people are generally treated much better by police; calm, no weapons brandished, etc., etc.
A shit-ton of black people have come forward recently, that they were doing something incredibly mundane and absolutely not illegal, when all of a sudden a police officer approached them (or pulled them over) and whipped out their gun, getting incredibly hostile and borderline abusive, all the while the black person was cooperating and not doing anything wrong.
THIS is the backbone of the BLM movement; to STOP this BS that occurs, simply due to colour of skin. Now, I know all of the stats about resisting arrest and all that, but it shouldn't matter; action should only be taken once things get to that point: de-escalate the situation, don't come out with guns a-blazing. You know, like how white people are treated.
Thinking about all of this, the absolute best way (in a non-brutal fashion, of course) this has been portrayed in mainstream media, as I can recall, is in Harold and Kumar go the White Castle.
Again, it's taken for laughs and it doesn't show just how far cops will go in real life, but that scene shows (and explains) how black people are perceived and treated, simply due to the colour of their skin.
TL;DR, BLM is strictly calling for blacks to be treated like any other, by policeman and not like animals or armed criminals.