You only need a cursory understanding of imaginary and complex numbers to be able to understand the Discrete Fourier Transform.1 While it is true that the output of the DFT is always given in complex numbers, and the formula itself contains a particularly abstruse bit involving the imaginary number i,
my advice is to not let the presence of i, e, and the complex numbers discourage you. In the next two sections we’ll reacquaint ourselves with imaginary and complex numbers, and see that the exponentiated e is simply an interesting mathematical shorthand for referring to our two familiar friends, the sine and cosine wave. I’m not trying to say that Euler’s formula is just some wacky bit of syntactic sugar. However, for someone who just wants to understand how the DFT works, it’s really not crucially important to have a deep understanding of Euler’s formula.
After that, I suggest listening to this broadcast of In Our Time where an enthusiastic Marcus Du Sautoy will guide you through the history of imaginary numbers. You'll learn all about Gauss, and the two-dimensional representation of complex numbers that was discovered by Gauss and Argand. It's really fun!