Be sure you have selected a LIVE profile or an Offline profile before starting a new campaign. As the game states, if you have not selected a LIVE profile or an Offline profile to play with and simply close the Games for Windows LIVE menu, you will not be able to save your game's progress.
Follow the instructions to save your game:
1. Launch the game.
2. Choose a profile.
3. Select New Campaign from the Main Menu.
4. Select your desired Difficulty level.
5. Select Yes when prompted to create a save file.
After completing a level, a message will be displayed indicating that the game is being saved, then the level stats will appear.
When the game is re-launched and the same profile is selected, the Main Menu will offer two additional options: Continue and Replay. Choose Continue to pick up the story from your last save point.
Where are the saved games located for STAR WARS™ The Clone Wars Republic Heroes for PC?
You can find them under C:\Documents and Settings\All Users\Application Data\SWTCWRH.