Minimal reproduction (in VS 15 Preview 2):
declare var foo: any;
Unable to activate Windows Store app [...]. The wwahost.exe process started, but the activation request failed with error 'Windows was unable to communicate with the target application. This usually indicates that the target application's process aborted. More information may be available in the Debug pane of the Output window (Debug->Window->Output)'.
There is a work-around for VS 2015 Update 3 (s/o to @minestarks on GitHub):
I assume you're running in an English locale here, otherwise I don't think the workaround is applicable.
Close VS. In an administrator command prompt:
cd %ProgramFiles(x86)%\msbuild\microsoft\visualstudio\v14.0\typescript mkdir en copy *.xaml en copy TypeScript.Tasks.dll en\TypeScript.Tasks.resources.dll
Now open your UWP project again, the TS files should be visible and building when you build your project.