The rms current of a square-wave signal is obtained by multiplying the peak value by the square root of the duty ratio \$D\$ as shown in the below picture:
The average value is the peak multiplied by the duty ratio \$D\$.
The rms value is the one you should use to size the fuse you need with some margin of course. This is all about power dissipation linked with the fuse resistance \$R\$.
I think the term averaged creates the confusion in the text. It does not refer to an average current but more to the final rms value seen by the fuse, in steady-state if you wish, once the converter or the equipment has started up and is stabilized. That would be my interpretation at least.