Basically to prove that something existed before a specific date. For instance, grab a random piece of paper on your desk with writings and then try and prove to me that the contents of that paper existed before last year, you can't really do it (easily). Before this, the only (easy) way to achieve this would be to send this document to a third party to store into a database that it did infact exist when sent to the database, but how can you prove that database wasn't manipulated? Another common way to prove something existed before a certain time would be to make a copy and send the copy through the mail to yourself. One you have received this copy from the mail, leave it unopened and now you have proof (through the mail service's date on the stamp) but this isn't too easy, some what of a hassle, and I think you could debate that you could fake it. (Faking the stamp, finding a way to get an opened document through the mail and than replacing the content later, etc.)
What this does is it stores a document's signature into the actual "history" of bitcoins which is public. ANYONE and EVERYONE can access this history of all transactions and by them storing the actual signature of the document into this public "log" you are now able to say that this document DID exist before the specific bitcoin transaction. This removes the hassle of all current methods of proving for existence and now no longer rely on a single third party (which could manipulate it themselves).