Why is Apprentice launched after installing the Free Engine plugin for Unity/Unreal?

The free Houdini license is for running Houdini Engine for Unreal and Unity, not Houdini.

Houdini Engine is a non-graphical version of Houdini FX. It does not allow you to open the Houdini interface. It will let you take existing (pre-made) Houdini Digital Assets, (you can get these at Orbolt.com), and load them into UE4 or Unity.

You can learn more about how Houdini Engine for Unreal and Unity works here.

If you try and open the Houdini interface without purchasing the Houdini FX or Houdini Core license, it will prompt you to install the free Houdini-Apprentice license.