In the "How to Configure Syslog on Cisco Router" YouTube video, the speaker demonstrates how to set up syslog on a Cisco router to store messages generated by the router, including debugging messages, on a syslog server. The speaker explains the importance of managing debugging messages to prevent network disruption and overloading the CPU. To configure syslog, the speaker shows how to turn off debugging, assign an IP address to the syslog server, and update the router's clock. The speaker then demonstrates configuring logging traps to log specific message types and setting the severity level for each message. The syslog server records messages with time stamps, which can be checked by ensuring commands are entered without the "no" parameter.