How can I comprehend the complex coefficient of complex Fourier series?
if you think of $e^{j n \omega_0 t}$ as a "vector" (complex numbers are not quite the same as 2-dimensional vectors, but they look the same, and are both describable with two real numbers) with length 1 and starting out with an angle of 0 (pointing straight to the right, along the positive real axis, and this vector is rotating counter-clockwise at a rate of $n \omega_0$ radians per unit time.  this vector is what you see looking down the spiral in the graphic referred to in the question, looking along the time axis.

then multiplying by $C_n$ changes the length of the vector to $|C_n|$ and the initial angle of the vector is the angle of $C_n$ or $\arg\{C_n\}$.

that's how i would visualize it.