The language, also known as Akkadian, was reconstructed by a group of scholars from Cambridge University headed by Dr. Martin Worthington. In order to uncover the secrets of the language's sounds, the group studied letter combinations and patterns internal to the written language and compared it to transcriptions into other, better known ancient languages.
Why does it matter that we can now have a reasonably accurate idea of how the Babylonian language sounded? Aside from possibly shedding light on the past in ways we can't anticipate, it ties us to our own past. Just Hammurabi's laws and the Gilgamesh story alone are high points in human response to a chaotic world. Not to mention that one might easily make a case for the transfer of information from an aural to a written speech system and back at will is the genesis of computer science.
And also, come on. How cool is it? It is very cool!