EDIT
Why is this issue extremely painful? So far while I debugged/tested my Web App I've just edited the .ts in VS IDE, saved, then reloaded the page and tested the changes. Now I have to build. However build is not enabled when debug session is running in the VS IDE. So now I have to do a complete roundtrip: edit .ts -> save -> stop debugging -> build -> launch debug. Again, this is extremely painful. END EDIT
I've unchecked / saved / checked the setting in TypeScript Build tab in my project setting.
The appropriate <PropertyGroup> was generated and saved to my .csproj file.
(note without changing any TypeScript Build setting, this <PropertyGroup> is not there (when using an existing project what was previously created with VS 2013), and the build defaults are not the same as the defaults on the IDE UI so you are misleaded.)
Anyway after this change / save activity my .csproj file is in sync with the IDE UI here it is:
<PropertyGroup Condition="'$(Configuration)' == 'Debug'">
<TypeScriptRemoveComments>True</TypeScriptRemoveComments>
<TypeScriptTarget>ES5</TypeScriptTarget>
<TypeScriptCompileOnSaveEnabled>True</TypeScriptCompileOnSaveEnabled>
<TypeScriptNoImplicitAny>False</TypeScriptNoImplicitAny>
<TypeScriptModuleKind>None</TypeScriptModuleKind>
<TypeScriptOutFile />
<TypeScriptOutDir />
<TypeScriptGeneratesDeclarations>False</TypeScriptGeneratesDeclarations>
<TypeScriptNoEmitOnError>True</TypeScriptNoEmitOnError>
<TypeScriptSourceMap>True</TypeScriptSourceMap>
<TypeScriptMapRoot />
<TypeScriptSourceRoot />
</PropertyGroup>
As it shows the Compile on Save is definitely enabled. Still the .js file only generated during build and not when the .ts file was saved.
Btw: Regarding that TypeScript compile is an msbuild task in VS 2015 implementation it is interesting how could this Compile on save setting even work...
More diagnostics: Interestingly in case if I delete the whole propertygroup then it actually compiles on save. (The UI remains the same, and the .map files are not generated on the save, only on build.)

