I know sills rusting is a problem but I don't understand why/how.
do I need to clean inside before treating?

 

My 3-cent worth to that topic:
You live in GB. I assume that the UK flood the road with salt in the winter - just like Germany? So this salt will accumulate on the backside of the sills and salt and water/humidity are the perfect environment to accelerate corrosion. This problem is not unique to the S-Type, but it affects all cars (as frequently shown in many episodes of "Car SOS").

If I had an S-Type in the UK (I do not have that issue with salt on the road in Australia), I would try to get access to the sills from behind, clean the backside properly, then clean it with wax&grease remover, and then slab a thick layer of "underbody protection" (=Unterbodenschutz) on there. This is some kind of black tar-like paint - not nice to look at, but that does not matter, if you apply it only where it cannot be seen...

 

As Peter says, salt and dirt. Wheel arches are hard to clean so few do it well and where this stuff collects stays damp longer and corrodes faster - poultice corrosion.Hose the salt away after each bad-weather run is helpful too.

Mine hasn't been used on salted roads and it is in good condition, so probably proof that the salt and muck are the culprits..