Installing Reign of Giants (Standalone) on PC
If you bought Reign of Giants from Humble Store, you should have received an email that contains a link to your Humble download page. Here, you should be able to choose your OS and download Reign of Giants for your standalone.
Once you've downloaded the installer for your Reign of Giants (on standalone), double-click and install the expansion into the same folder as Don't Starve. For example: C:\Program Files (x86)\DontStarve\
Once installed, when you start Don't Starve, you should have the Reign of Giants content accessible whenever you start a new game world.
Installing Reign of Giants (Standalone) on Mac OS X
If you picked up Reign of Giants from Humble Store for your Mac OS X, you should have received an email that contains a link to your Humble download page. There you should be able to choose your platform/OS and download Reign of Giants for your standalone.
This will download the .DMG for your Mac OS X. Double-click the .DMG and it should open a new window with the Reign of Giants logo in it. Drag that into your applications folder and it should create a Reign of Giants installer.
For Older OSX Versions: Double-click the installer and ensure that it installs Reign of Giants in the same location as your original Don't Starve files.
For Newer OSX Versions: Control + Click the app, select "Open" from the dropdown menu, OSX will ask you if you would really want to open the file, click "OK".
When you start Don't Starve, you should have the Reign of Giants content accessible whenever you start a new game world.
Installing Reign of Giants (Standalone) on Linux
If you purchased Reign of Giants from Humble Store for your Linux OS, download your version of the Reign of Giants installer (32-bit or 64-bit).
Once the zipped files have been downloaded, extract the contents into the same folder as Don't Starve. When you start Don't Starve, you should have the Reign of Giants content accessible whenever you start a new game world.