Extensions are run by a Ruby interpreter that is embedded within the SketchUp process itself. There is no separate process you can monitor. You can, however, use the sketchup Ruby debugger interface to control the execution of an extension from various code development suites.
I don’t know that the developers have ever publicly documented the details. The dribs and drabs I’ve seen here suggest that some of SketchUp’s dialogs are implemented as web pages, and the sketchup_webhelper.exe acts like a sort of local server to support them.