my favourite intuition about Euler, which is not really an explanation, somewhat tautological, and may or may not be wildly incorrect, but I like it nonetheless:

e^x is a function whose value is its rate of change. (De^x=e^x). Now imagine the unit circle by taking a point an unit away from O, and set "rate of change" perpendicular to that vector. You will end up with Df(x) = i f(x), which really only works when f(x) = e^ix, supposing i means perpendicularity.