How do I explain a complex exponential intuitively?
From a graphical point-of view, it is  an infinite spring, whose distance between adjacent coils reflects the frequency of the complex exponential:



If you have a 1D time x-axis, you may be used to draw functions along a single 2nd y-axis  dimension: sines, cosines, etc. If you want to plot a complex function,  you need one x-axis, and 2 y-axes for the real and imaginary parts. You can plot this into a 3D visualization, see above, and you see a spring (although it is some hot summer here). Changing the frequency dilates or contracts the spring. It is known as the Heyser corkscrew, or Heyser spiral, in DSP. 

A little more details can be found there:


3D wiggle plot for an analytic signal: Heyser corkscrew/spiral
DSP Concepts Visually Explained


The Fourier transform tells you that any wire (a function) can be reproduced by a superposition of scaled and shifted springs.

After the what, the why. Complex exponentials (or cisoids) are special in  that if one is filtered (with a moving average) it keeps the same shape. So, they are invariant, under  Linear-Time-Invariant (LTI) systems. Invariant vectors/functions are often an appropriate way to study systems or transformations. Morever, as they form an orthogonal basis, they form a basis of choice to decompose arbitrary vectors, and to study how the latter are affected by LTI systems.

Last, the complex exponential is itself invariant under differentiation ($(e^{z})' = e^{z}$), a specific linear and invariant operator), which makes it quite unique, with interesting properties.