The box is just a textarea within a form:
Replace the .....s with parameters, like the action of the form and the styling.
<textarea .....>any text you want goes here including any html tags</textarea>
I'm not sure if you want more info because the site you linked to is quite complex. It has a form, yes, but you would also need to generate the result for that. Just having the form doesn't need any server side code, though.
Also, freewebs doesn't support PHP, etc., so you won't be able to make a generator. You can, though, have a textbox from which you can cut/paste code.
Note: the reason it doesn't "save the original page" is that it is a PHP script that checks if the form has been sent (and if the info is valid). If so, it generates the preview and the textboxes. If it wasn't sent, then you get no form displayed.