We don't do drafts anymore, and I'm not saying it's because of him but the fact that he won his case in the Supreme Court with a 8-0 decision, I think he had an impact. If nothing else, religious freedom.
He also used his platform as the most famous athlete in the 60's to call out the treatment of minorities, militantly so in a lot of cases (which is why I said you didn't have to like him, my grandfather hated the guy and told me every chance he got).
Like I said, you don't have to like what he said, but to the question of "why should I care and why is it such a big deal that everyone has made a statement at some point on my television?" It's that he was a polarizing figure whose legacy outside of sports is just as noteworthyl as being the greatest fighter of all time. There aren't many people who will get as much attention when they die for that reason.