To begin with, you've got a false narrative of the Trayvon situation. Martin's family's attorney said he didn't think the case was really about race. The neighborhood where it happened was 20% black, and of course Zimmerman was Hispanic. But the immediate impulse of many was to make the entire thing about race, with the event being described as a black boy being shot to death for going to buy skittles and walking back to his home in a neighborhood where he stuck out, in a hoodie. Even though the jury found Zimmerman not guilty, and even though the evidence suggests Martin was the aggressor, various luminaries treated the situation as emblematic of black people being victimized by whites, and some people still endorse that view.
Initial reports made it sound like Zimmerman had stalked a random black kid, provoked a confrontation, and shot him. A picture of Martin from years before where he looked very young was published, which obscured the fact that Martin was much larger than Zimmerman. What actually happened is that Zimmerman called a non-emergency number to report Martin looking in windows of houses after a neighborhood watch had been started to deal with burglaries. Zimmerman watched Martin until Martin was out of sight, then that's when Zimmerman was told by police that he "didn't need to" follow Martin (Zimmerman replied, "OK"). Zimmerman did get out of his car, but he didn't find Martin. Martin then ambushed Zimmerman, got on top of him and banged his head into the ground. That's when Zimmerman shot him.
So what was the purpose of the narrative that Zimmerman was a white person who attacked a young black kid just for being black? Why do some people just "know" that's what happened? Did Obama's remarks contribute to the perception that black people were under attack from whites?
This has to do with BLM because, rather than being emblematic of black people being victimized by others for being black, it's much more emblematic of a false narrative being propagated based on murky details, at best, and fantasies, at worst. Black people are not disproportionately targeted for violence by non-blacks, and in fact black people are disproportionately more likely to be the perpetrators of violence against non-blacks (of course, the vast, vast majority of violence is intraracial). It's questionable whether black people are more likely to be shot by police relative to crimes committed by blacks. But here we are, and this stuff really did start under Obama with Trayvon and Michael Brown.