The principle is simple; a flaw that uses forged UDP packets is to be used to trigger a rush of DNS queries all redirected and reflected to those 13 IPs. The flaw is as follow; since the UDP protocol allows it, we can change the source IP of the sender to our target, thus spoofing the source of the DNS query.
The DNS server will then respond to that query by sending the answer to the spoofed IP. Since the answer is always bigger than the query, the DNS answers will then flood the target ip. It is called an amplified because we can use small packets to generate large traffic. It is called reflective because we will not send the queries to the root name servers, instead, we will use a list of known vulnerable DNS servers which will attack the root servers for us.