Depending on your OS:

Vista:
C:\Users\username\AppData\Local\Ironclad Games\Sins of a Solar Empire\

XP:
C:\Documents and Settings\username\Local Settings\Application Data\Ironclad Games\Sins of a Solar Empire\

Please note these directories may be Hidden by default. To unhide them, go to your Control Panel -> Folder Options -> View and set yourself to view hidden files and folders.

That path contains folders to put replays in (AutoRecord-Singleplayer and AutoRecord-Multiplayer). So throw it in there, then in-game go to single player > watch.