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 Either the graphics card's drivers are out-of-date, or because it does not have sufficient dedicated memory (256MB) to render Don't Starve or Don't Starve Together.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 your game cache 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 Out of date drivers can cause issues with software.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.
If you have tried all troubleshooting steps above and are still encountering the issue, please contact our support team with your DxDiag.txt/system specs and your client_log.txt files.