As part of our agreement, GoG.com has specifically decided not to provide Steam keys. With your Don't Starve purchase on GoG.com, you do have access to download the standalone version of the game. With this version, you'll have access to community-created mods and offline play.
Once you run it for the first time, it will ask for your game key, enter your game key and click on the "Next" button and your game should update to the latest version!
If you have any issues with updating the game, you'll find an easy-to-follow walkthrough here: How to install and update your DRM-free client
