How does TAB auto-complete find options to complete?

Depending on the command:

• Someone may have written a function to generate possible completions of arguments, including options. You'll find functions for some commands in /etc/bash_completion.d/* (or a different location on some systems). These functions are registered with the complete built-in (e.g. complete -F _find find tells bash to call the _find function when you press Tab on a find command). They use the compgen built-in to tell bash “here are the possible completions”.
• For some commands, bash will call the command with the argument --help and parse the output. Such commands can be registered with the complete built-in, e.g. complete -F _longopt ls. _longopt is in fact a completion generation function, that happens to parse a command's output rather than use a fixed list. (There are other more specialized completion functions that parse a command's output to generate possible completions; look in /etc/bash_completion.d/* for examples.)
• For things like aliases, the completion function looks them up in bash's internal tables. The complete built-in has options for that, e.g. -A for aliases.

Have a look at the file

/etc/bash_completion

and observe the files from the directory:

/etc/bash_completion.d

You will find the answer.

Support for bash completion is provided in Debian and Ubuntu by the bash completion package. You also usually need to uncomment the following in /etc/bash.bashrc and/or .bashrc to source the bash completion files. The following is from /etc/bash.bashrc:

# enable bash completion in interactive shells
if [ -f /etc/bash_completion ] && ! shopt -oq posix; then
    . /etc/bash_completion
fi

for this to work. I usually uncomment it in both. The handling of interactive/login shells in Debian is a bit of a mess.

Similar comments presumably apply to other distributions, though maybe they enable the sourcing by default.