I think this feature supports three desirable aspects of the site:
Linkability: An article should provide a link where it can be found. This can have additional benefits such as being a permalink or a link without any navigational parameters, as pointed out in other answers. Ultimately, though, it is a way for the reader to determine the URL of a page (for finding the up-to-date version again, for passing on the link to others, ...) as opposed to the current URL of the file they might be reading.
If the article is read in its original location, the browser's address bar may serve as an alternative way of retrieving that URL. However, relying on the address bar is a design mistake IMHO - for all we know, the page might be displayed in a frame/iframe of sorts, or the browser might be running in some kind of weird kiosk mode that does not allow the user to access the address bar. Moreover, all the other URLs to in-article resources (e.g. on Stack Exchange, links to single answers or comments) can be obtained by finding the appropriate links on the page, so getting the article's own URL via a link, too, seems fitting.
Testability (of links): My main point above has been retrieval of the URL. Now, there would be several ways to do that - the URL might also be written verbatim or in a text box, ready for copy-and-pasting. However, by providing an actually clickable link, the URL that users might copy via the context menu is testable right away. It only takes a single click to verify the link really does point to the desired resource, before copying that URL to some other destination.
Consistency: If the title of an article is a link to the article on some pages of a website, it surely should be a link to that article on all pages of that website. Anything else would be inconsistent and thus confusing.