What you have come across is the yet to be fully explored world of cryptographic signatures - and this is only the beginning my friend.
A cryptographic signature in a very simplistic way, allows you to electronically sign something, usually a string of text. That signature can then be used to prove that the text that you signed has not been tampered with. If you were to even add a comma or a space to the text, the outcome of your signature would be different, and therefore you could prove that the text and your signature were not the same.
Unless you did not know, almost anything digital is at one level a string of text (sometimes human readable, sometimes not) so anything digital can be cryptographically signed in one way or another.
This has absolutely nothing to do with the block-chain or bitcoin per se, but is a side-benefit of bitcoin employing crytographic signatures to allow you to send bitcoins. You do not have to spend bitcoins to sign something.
In short, the mechanism is being used to allow you to crytpographically sign ANYTHING. It is almost too broad a subject to answer fully here, but for example, imagine that you are buying a house, and that you copy paste the contract into the message box and sign it, and then the seller does the same with the result of your signature.
What you would get is a cryptographically signed contract that absolutely could not be tampered with, that proves that both of you agreed to the terms in the contract, and an independent third-party could prove this to be the case.
This opens up a whole new world potentially without lawyers (think of how wondeful that might be) because you would only agree to (and sign) things that you agreed 100% with and understood. You could not be forced to sign anything that you did not agree with.
In the future where cryptographic signatures are used for everything people and companies will be forced to behave fairly in order to get anywhere at all.