The drive connects to your local network via an ethernet connection on your router. If you want to access the drive outside your local network (over the Internet), you will need an Internet connection.
yes, you connect to the drive through a gigabit ethernet connection (which should backwards-compat to a 100M network if you have that). I have a gigabit LAN here, but it should also work if you have a lower-speed network. So it's a LAN connection, which technically should not need an internet connection.