Why when announcing a King's death did people say Long Live the King?

Because of something called the royal secession. Meaning that as being King was quite important, people kept close tabs on who would become the next king if the current king should die. Much like the president , and the vice-president only that the chain goes on much longer. And even though the actual coronation could keep for a few days, the moment the old king dies, the power immediately goes over to the next guy in line.

So it's basically saying Goodbye Old King, and welcome New King.