The octothorpe/number-sign/hashmark has a special significance in an URL, it normally identifies the name of a section of a document. The precise term is that the text following the hash is the anchor portion of an URL. If you use Wikipedia, you will see that most pages have a table of contents and you can jump to sections within the document with an anchor, such as:
https://en.wikipedia.org/wiki/Alan_Turing#Early_computers_and_the_Turing_test
https://en.wikipedia.org/wiki/Alan_Turing identifies the page and Early_computers_and_the_Turing_test is the anchor. The reason that Facebook and other Javascript-driven applications (like my own Wood & Stones) use anchors is that they want to make pages bookmarkable (as suggested by a comment on that answer) or support the back button without reloading the entire page from the server.
In order to support bookmarking and the back button, you need to change the URL. However, if you change the page portion (with something like window.location = 'http://raganwald.com';) to a different URL or without specifying an anchor, the browser will load the entire page from the URL. Try this in Firebug or Safari's Javascript console. Load http://minimal-github.gilesb.com/raganwald. Now in the Javascript console, type:
window.location = 'http://minimal-github.gilesb.com/raganwald';
You will see the page refresh from the server. Now type:
window.location = 'http://minimal-github.gilesb.com/raganwald#try_this';
Aha! No page refresh! Type:
window.location = 'http://minimal-github.gilesb.com/raganwald#and_this';
Still no refresh. Use the back button to see that these URLs are in the browser history. The browser notices that we are on the same page but just changing the anchor, so it doesn't reload. Thanks to this behaviour, we can have a single Javascript application that appears to the browser to be on one 'page' but to have many bookmarkable sections that respect the back button. The application must change the anchor when a user enters different 'states', and likewise if a user uses the back button or a bookmark or a link to load the application with an anchor included, the application must restore the appropriate state.
So there you have it: Anchors provide Javascript programmers with a mechanism for making bookmarkable, indexable, and back-button-friendly applications. This technique has a name: It is a Single Page Interface.
p.s. There is a fourth benefit to this technique: Loading page content through AJAX and then injecting it into the current DOM can be much faster than loading a new page. In addition to the speed increase, further tricks like loading certain portions in the background can be performed under the programmer's control.
p.p.s. Given all of that, the 'bang' or exclamation mark is a further hint to Google's web crawler that the exact same page can be loaded from the server at a slightly different URL. See Ajax Crawling. Another technique is to make each link point to a server-accessible URL and then use unobtrusive Javascript to change it into an SPI with an anchor.
Here's the key link again: The Single Page Interface Manifesto