Don't Stave Together Won't Open...at all

The error message you are seeing is because your computer does not have a specific DirectX .dll file from Microsoft. Here are a couple of things you can try:

Re-install the executable file called DXSetup in your Steam folders for Don't Starve Together:
C:\Progarm FIles\Steam\steamapps\common\Don't Starve Together\DXRedist\DXSETUP

Re-install VCRedist in the Don't Starve Together folder:
C:\Program FIles\Steam\steamapps\common\Don't Starve Together\VCRedist

Should the issue persist after trying the steps above, please visit https://www.microsoft.com/en-us/download/confirmation.aspx?id=8109 to download the DirectX End-User Runtimes from Microsoft's official website, install the file, and try launching the game once more.