Does anyone know how a stormscope is able to sense the distance to the
lightning ...

The booklet that came with ours used the term "pseudo range." Ryan
(the inventor) measured the strength (amplitude) of a number of
lightning strokes, and calculated the average.

The stormscope assumes that every lightning strike is of "average"
intensity, and it calculates the distance from the signal strength.

A strike that is stronger than average will seem to be closer than it
actually is; a weak one will seem to be farther away. Hence "pseudo."

But it's good enough to provide safe guidance to the pilot.

vince norris