By default, KSystemLog will attempt to open itself with elevated privileges. This, in turn, allows it to not only display logs but also manage the processes that are currently running in the system.
Because of that, KSystemLog will always try to run itself as the root user when you open it for the first time. In order to achieve that, however, it first needs to know your root account's password.
Despite that, it is still possible to use KSystemLog without providing any root privileges. To do that, you can press the "Ignore" button when KSystemLog asks for any elevated privileges.