It's a common figure of speech. A drumbeat is a steady pattern of loud, attention-getting sounds. Here, they are talking about a series of news events that appear to be coming almost regularly and that are forcing Americans to start paying attention. He could have said "recently, Americans have started to notice that shootings of unarmed black men by police are occurring regularly." By using "drumbeat" as a figure of speech, he ties it in with the idea of "awakening." The shootings have been going on for a long time, but only recently have Americans woken up to them. "Drumbeat" can also mean a rising tide of publicity or public opinion. It would be common to say something like "we are starting to hear a steady drumbeat of publicity for X..." "Drumbeat" carries the idea of "something that calls us to action," as when drumming was used as a signal on British naval ships in the days of sail.