ideally, it should open automatically with a media player of some sort.
on the listen button, right click and then click on 'save target as'. select a place to download the file, and open it. it should open automatically by default with windows media player. if not, right click the file again > open with > and then select windows media player/winamp/itunes or whatever you have installed on your computer.
have you got a firewall installed? if so, check the settings or turn off the firewall and try liveatc.net again. if it works, then you've isolated the problem.
you could try downloading and using firefox your internet browser as it could also be a problem with internet explorer.
http://www.mozilla.com/en-US/firefox/