Isn't it insane how close the race between G.W. Bush and Al Gore was in 2000? Consider the consequences of that decision for a moment. And the voter turnout was only 51.2% in that election, not ~70%. Saying that in the end a small amount of voters make all the difference, that's true in so many cases...
At least regarding the Brexit-vote no one can complain that there wasn't a discussion about the issues, everyone spoke out, politicians, musicians, actors, companies, banks, ... and the people were bombarded with information and arguments for months. And the people have spoken.
(Others have pointed out that this is not legally binding, true, but it's certainly going to happen. There is no other viable way at this point)