The "error during initialization" error message is most commonly seen when the graphics card is not able to render the game. Either the graphics card's drivers are out-of-date, or because it does not have sufficient dedicated memory (256MB) to render Don't Starve or Don't Starve Together.
Here are a few things you can try to resolve this issue:
1) Try verifying the integrity of game files; or uninstalling, then re-installing the game to see if that helps. (https://support.steampowered.com/kb_article.php?ref=2037-QEUH-3335)
2) Ensure that all of your system's drivers (graphics card, audio card, operating system, etc...) are up-to-date. Out of date drivers may cause various issues with software.
3) If you have a dedicated graphics card like an AMD Radeon or NVIDIA GeForce card installed, you will have to go into your AMD or NVIDIA Control Panel and force your dedicated GPU to launch and run Don't Starve or Don't Starve Together instead of your integrated card.
4) Update your Windows operating system to install the latest updates/patches for the operating system via the Windows Update tool.
If you have tried all troubleshooting steps above and are still encountering the issue, try installing the DirectX Dependency Fixer (DXFix) tool. (Please note that this software was not developed or supported by Klei Entertainment and you must use it at your own risk.)
Should all steps not help, please contact our support team with your DxDiag.txt/system specs and your client_log.txt files.