The story line of the game takes place twenty years after a few powerful heroes saved Sanctuary from the demonic onslaught in Diablo II. This has gone by largely unnoticed by the general population as most of those warriors who directly faced Hell's armies (and were fortunate enough to survive) went mad from their experiences. Some people know that some sort of conflict took place, but for the most part the war against Diablo is a myth.
What happened to Tyrael after he destroyed the World Stone is unknown to everyone. Scholars like Deckard Cain anticipated a huge invasion of demonic forces as the protection from the High Heavens and the Burning Hells, but no invasion have begun. Cain has spent the last years trying to figure out what happened, and trying to prepare us for the inevitable onslaught.
We know from the Diablo III cinematic that a great comet of some importance is falling down on to the Tristram Cathedral, and that some unlucky soul (if it's Leah or Cain isn't sure) is hit by it. Hell's armies are amassing, and it isn't safe any more.