Why and how am I supposed to change the extrusion multiplier?
In an ideal world, you will not need to change this parameter once it is properly calibrated.

In a non-ideal world, some filament may be out-of-spec, and some filament may slip when it passes through the extruder. So for a flex (or more flexible) filament, you might increase the extrusion multiplier a little to compensate. If the compensation is material specific, it should be consistent (and might even be advised by the filament manufacturer).

You might want to increase/reduce the multiplier whilst printing the first layer (in lieu of fixing your bed leveling properly).

Maybe you want to fine-tune the top surface (which will be 100% fill) and you prefer to slightly under-fill this (because over-fill results in more noticeable surface defects).

It is for quick hacks or fine tuning, there is probably no scientific approach to setting any value other than 100%.

When it comes to material specific variations (due to melt viscosity, thermal expansion or drive efficiency), there are many factors which affect the tuning. Machine geometry, temperatures, colourants (and other additives in the filament), as well as the base material.