What is most likely happening is the game (or the Steam Client) is not being run with the proper Windows User Account administrative permissions - which is needed in order for the game to read/write files on your hard drive.
Ensure that your Windows User Account has the proper administrative permissions to run programs and apps. Please ensure that you are running the game in administrator mode if possible. Please visit Microsoft Support's website for information on ensuring administrative permissions: https://support.microsoft.com/contactus/
1) Run the Game as an Administrator
- Try running Don't Starve dontstarve_steam.exe as an administrator in your game folders (Steam\steamapps\common\Don't Starve Together\bin\dontstarve_steam.exe)
- Right-click on it and select "Run as Administrator"
- If that does not work, go to step #2.
2) Run Steam as an Administrator
- Fully close out of Steam and restart your computer.
- After restarting your computer, right-click on the Steam icon > select "Run as Administrator"
- Try launching the game after you have administrator access.
If Steps #1 and Step #2 do not help with the issue, try Step #3 and these should be able to help resolve the issue.
3) Check your Anti-virus, anti-malware, Firewall, and security settings
- Depending on your firewall, security, and antivirus settings, occasionally it may block a program that you want to access. Ensure that your firewall, security settings, and other antivirus software is allowing the game to do so. You can find more information here: https://support.steampowered.com/kb_article.php?ref=9828-SFLZ-9289