how to write value of variable into current edit file in vim script

How about the following?

execute "normal! i" . TodayDate

This will put you in insert mode and insert the content of TodayDate at the cursor position.

You can use :put to put the contents of the variable (or expression) into the current buffer

:put =TodayDate

The help for :h :put

                                                        :pu :put
:[line]pu[t] [x]        Put the text [from register x] after [line] (default
                        current line).  This always works linewise, thus
                        this command can be used to put a yanked block as new
                        lines.
                        The cursor is left on the first non-blank in the last
                        new line.
                        The register can also be '=' followed by an optional
                        expression.  The expression continues until the end of
                        the command.  You need to escape the '|' and '"'
                        characters to prevent them from terminating the
                        command.  Example: 
                                :put ='path' . \",/test\"
                        If there is no expression after '=', Vim uses the
                        previous expression.  You can see it with ":dis =".

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For mappings and editing <C-R>= is probably better than :put since it allows you to use the expression register and output the contents at the cursor location. (Take a look at :h <C-R>)