Debian-installer main menu has an "Execute a shell" option which starts an ash(Almquist shell) in BusyBox environment. One can exit this environment back to debian-installer with the exit command. What are the practical reasons for such BusyBox environment? Or does the debian-installer actually run inside this very same BusyBox environment?
The installer actually runs inside that environment. You can also get a shell by switching to virtual console 2 or 3.
Having a shell available on the install media turns out to be really useful to use it as a rescue disc or otherwise fix up special weirdness needed to make the install work (as an example, see my answer about LVM RAID, in particular the how to install section).
BusyBox is a set of basic user and administration utilities including
mount, etc. The Debian installer uses some of these utilities for its own purposes because it needs to perform these tasks.
In addition, there is an option for the user to get a shell prompt and run commands. This is useful when you need to do something that the designers of the installer didn't predict, for example because you have hardware that isn't supported or because you have an unusual setup.
The installer comes with BusyBox rather than with the GNU utilities found on the final system because BusyBox is significantly smaller (at the cost of having fewer features). While this isn't as important nowadays as it was in the days of floppy disks, there are still many circumstances where it's useful to save a few megabytes in the installer image.