Since the new Windows Terminal is still in active development, it often tends to develop starting troubles, especially after Windows updates. Here are three possible solutions.
Make a backup of the settings.json (formerly profiles.json) file before attempting these solutions. The first solution does not require a backup but make one anyway. Settings.json can be found in %LocalAppData%\Packages\Microsoft.WindowsTerminal_8wekyb3d8bbwe\LocalState
Solution 1: Repair
Search for Windows Terminal in the Start Menu. Right-click on Windows Terminal and choose App Settings. This option is also available in Settings ➾ Apps ➾ Apps and features ➾ Windows Terminal ➾ Advanced options.
Scroll down and click the repair button.
Solution 2: New settings.json
Move the settings.json file out of %LocalAppData%\Packages\Microsoft.WindowsTerminal_8wekyb3d8bbwe\LocalState. Try starting the Windows Terminal, it will create a brand new settings.json file. If it works, you can try to copy the old settings back bit-by-bit to see where it breaks.
Solution 3: Reset
In the App Settings window which we opened earlier (see solution 1), click the reset button. It will delete all the data of the app. Use this option only if the other options do not work.