How does the exit command work on a Unix terminal?

man bash

  exit [n]
         [...]  A trap on EXIT is executed before the shell terminates.

Such traps are often used to clean up tmpfiles on exit, see https://stackoverflow.com/questions/687014/removing-created-temp-files-in-unexpected-bash-exit

Define an exit trap like this (for better testing in a new shell):

$ bash
$ trap "rm filetodelete" EXIT

Show defined EXIT trap:

$ trap -p EXIT
trap -- 'rm filetodelete' EXIT

Test:

$ exit
rm: cannot remove ‘filetodelete’: No such file or directory

Note that exit may be "called" implicitly too. So instead of exit you could have also triggered the trap by kill -HUP $$.

Well usually you would only see execution upon exiting a shell if you've manually configured this. But maybe one of the packages you've installed came with a bash exit shell script...

check;

~/.bash_logout

maybe you'll find a script call from there, it's an odd one...

The exit command is a special built-in command in shells. It has to be built-in as it needs to exit the shell process.

It exits the shell with the exit status provided if any or that of the last command otherwise.

Upon exiting, the shell will run the EXIT traps if any. See the output of trap (in Bourne-like shells) for the currently set ones.

With many shells, if the shell was invoked as a login shell (some systems/users configure terminal emulators to start a login shell), it will also run the code stored in special files like ~/.logout, ~/.zlogout, ~/.bash_logout and possibly corresponding ones in /etc depending on the shell.

You could do a set -x before calling exit to get an idea of where those commands are being run from.