Steam synchronizes many backup files on its servers. They are automatically downloaded from Steam when you install a game. However, this is not the only way to get it. You can also download them directly from the Valve website in your browser.
Enable Steam Cloud Syncing in Steam
If Steam does not automatically download your old games after installing a game, make sure that Steam Cloud is enabled for Game in Steam.
To do this, find the game in your Steam library, right-click on it and choose Properties. Click the "Updates" tab and make sure that the "Enable Steam Cloud Sync" option is turned on the game. If this option is not enabled, Steam will not automatically download or upload your cloud storage.
If a Steam Cloud option is not displayed here for a game, this game does not support Steam Cloud. Not all games on Steam can ̵
Download Files in Your Web Browser
Valve also lets you download your Steam Cloud storage files through a web browser. You can only download your backup files without downloading the entire game again.
To find your backup files, visit Valve's View Steam Cloud page in your web browser and sign in with your Steam account.
You will see a list of games using your Steam Cloud storage. Locate the game in the list (press Ctrl + F to use the browser search) and click Show Files to view all the files for a game.
Each game has a page with all the files stored in Steam Cloud and the date they were changed. To download a file, click "Download". Download all saved files for your game and receive a copy of the saved games.
This is especially useful for multiplayer platform games that do not automatically synchronize stored files between platforms.
Borderlands 2 is available for PC and Mac, for example. It syncs its backup files to Steam Cloud, but both the PC and Mac versions sync the backup files separately. You can download the Mac (or PC) versions from the Steam website and then manually import them into the correct folder to move your Borderlands 2 games between platforms.
Without using the website, this requires downloading Borderlands 2 on one platform, extracting your saved games and then downloading the game on a second platform!
This is also a known issue for other games, such as when stored games are to be moved from Torchlight for Windows to Torchlight for Linux.