Ping of Death attacks are things of the past. Operating systems no longer freak out when assembling large fragmented ICMP packets.
That video is trying to demonstrate a ping flooding attack, a DoS attack in which he's trying to send more ICMP packets than what the server can handle. Chances are that the server being "attacked" has bigger downlink that the attacker's uplink. Allow me to demonstrate.
The guy in the video is trying to do something like this:
As you can see, he'll never be able to flood the server's downlink/resources. For this attack to be successful you need something called a Distributed Denial of Service attack (DDoS), in which a large number of attackers try to flood the server's resources
To learn about mitigating DDoS attacks, you can start by checking this question about different techniques used by firewalls.