Double-click to run the Dedicated Server and choose "Run Pre-configured Servers" from the pop-up.
This will pop open two windows, one for the Overworld and one for the Cave. Congrats! Your servers are running!
If you don't get this pop-up, you will need to force an update: right click on Don't Starve Together Dedicate Server > Properties > Local Files > Verify Integrity of Tool Cache. It should work for a little bit, and then be ready to go.
But you'll probably want to customize them a little so...