Theoretically, if you don't close your browser, someone can sit down at your machine and click the back button to see information associated with whatever you were doing - that could be bank accounts, credit card info, personal information - whatever. Unless the site screwed up horribly in their disabling of your session, they can't actually look at anything you didn't look at in that session, or actually change anything, but just being able to see the information might be all they need.
The reason this happens is that browsers tend to cache pages - if you click the back button, it will often load the page from its cache, rather than download it all again.
Of course, it's entirely possible that your browser decided not to cache these particular pages - I seem to recall that at least IE will never cache HTTPS traffic, although I could be wrong - but by showing the message at logout time, then you can't really say you haven't been warned.