See the forum thread here: http://lhcathome2.cern.ch/vLHCathome/forum_thread.php?id=1681. In a nutshell:
You will need to have an open internet connection while running project tasks. You cannot run the tasks if not connected to the internet.
The task starts up a VM, then the VM runs a series of tasks that each run for about 60 to 90 minutes each. At the beginning of each task inside the VM, it downloads some data, crunches the numbers, and uploads the result, then the next VM task starts and downloads more data.