He didn't create a Test Project in the examples. The examples tell you to create class libraries. Your TicTacToe.UnitTests class will generate a TicTacToe.UnitTests.dll in your bin\Debug folder and will not run like an executable.
Download NUnit from here http://www.nunit.org/index.php?p=download
Once NUnit is installed, run it and create a new project. Save project. Then add an assembly (Project\Add Assembly...). Point to your TicTacToe.UnitTests.dll.
You should see the project hierarchy load to the left and you should be able to click the Run button to run the tests.