[Posting some images here for possible future reuse.]
A non-rigorous argument: when is large enough so that is small, is roughly (hand-waving) the point on the unit circle at arc length (and hence angle) :
So multiplication by roughly corresponds to rotation by angle . Multiplication by , which is multiplication by n times, roughly corresponds to rotation by angle . As , “roughly” becomes exact.
Animation for :
Image generated from Python-generated SVG files; code available if anyone wants.
In particular, once one accepts the fact/definition that (for instance, show that the function satisfies ), it is true that is a rotation by π, that is,