Those replays are official (they are in game), but they are like in beta stage.
Replays in Vista are located here:
C:\Users\(USERNAME)\Documents\My Games\Gas Powered Games\Demigod\replays\(PROFILENAME)
There is always last game replay, so if you want to save some good game replay, rename or move LastGame.DGReplay file, before start another game. Otherwise your replay will be overwrited by a new one.
To play replay, the simplest way is to move replay file to C drive, and run game with parameter:
"C:\Program Files (x86)\Demigod\bin\Demigod.exe" /replay C:\LastGame.DGReplay
You can speed replay up with "+" and slow it down with "-" button. You can use ~ to view HP of demigods.
But right now replays have many problems. Eg. I'm always kicked out of replay after ~10 minutes in, so I can watch only around 10 first minutes. Still, the replay mechanism is in game long time now, but it works really poorly.