Your connection to the Internet is up, but servers on the Internet which allow you to find the greatawakening.win server at address 63.21.56.102 (for example) are down, so you have this local static mapping (which is included on all machines) that tells your system what the IP addresses are for specific domains. Every system checks this file before sending a request to a DNS server, so if the DNS servers are down this will allow you to still connect to some important sites.

They can't take out the whole Internet which consists of a physical infrastructure and nodes residing on that infrastructure. However, everyone needs DNS because they can't remember IP addresses, and DNS is sort of a weak point in the whole Internet architecture because it has no security and everything relies on it, plus most of the public servers are in the hands of corporate entities. If they simply deleted the entry for greatawakening.win from the server that originally hosts the DNS then nobody could reach the site without an IP address.