Why have a smaller fuse on input & larger on output on a power supply
Fuses work on \$i^2t\$ heating.

It's the current which causes the fuse to melt. If input requires less current and output gives more current, you will have to use a smaller current rated fuse at the input and a larger current rated fuse on the output. This is the usual case with step-down or buck converters.

However, if you are working with a step-up or boost converter, these will draw higher current on input and deliver lesser current (with higher voltage) on the output. In this case, you will find a larger current rated fuse on input and smaller current rated fuse on the output.