Depending on where you purchased Don’t Starve, you can retrieve your game in the following ways:
Steam:
If you purchased Don't Starve through Steam, the game would have been automatically added to your Steam game library (as it normally would). Log in to the Steam account that you used to purchase the game, and you will find the game in your Steam library.
-----------------------------------------
Humble Store:
When you purchase Don't Starve through the Humble Store, you will be sent an email that contains a unique URL allowing you to claim your game.
Where is my email?
If you either A) mistype the email address or B) type in a different email address than you think you used to purchase, you might miss the purchase email confirmation.
If that happens, your best, next step is to check any possible email address (and spam box) to see if you can find the Humble Store email confirmation.
You can also check if you have purchased Don’t Starve from Humble Store by logging into your Humble Store account and navigating to your "Purchased Products" page, where you should be able to find your Steam key as well as a DRM-free client installer.
If you are having trouble with a Humble Bundle Store purchase, please contact Humble Bundle support.
-----------------------------------------
GOG.com:
If you purchased Don't Starve from GOG, log into your GOG account and check the Account Orders & Settings page to find your order history and the game. https://support.gog.com/hc/en-us/articles/212184469
-----------------------------------------
Chrome Web Store: (During the Don't Starve Early Beta)
If you purchased Don't Starve through the Chrome app during the early beta and were not able to translate your Chrome key into a Steam key, please contact customer support with your Google Order Number and the email address you used to purchase the game.