What is the "Remove trailing whitespace button" all about?

You won't really be able to see any difference when viewing the page, only when you go back to edit it. To test it, edit a page, add some text with extra spaces at the end, like "hi there! " (don't include the quotes of course). Select the checkbox and save it. Go back to edit the page and you'll see that the trailing whitespace is not there, i.e. you'll only have "hi there!"