This is more math than physics, but I have sketched some quick diagrams that hopefully can give some intuition to this. Suppose we observe a set 9 discrete measured quantities as follows:
It looks periodic, but we do not yet know in what ways and how periodic it really is.
The main idea is to hit the observed series of values (ie. multiply the numbers point by point, as shown in examples below) with different sine functions, which we know are also periodic. We attempt to hit it with 3 different sine function as follows:
We see that only a very special function, the one in the middle, which actually has the same original shape as the observed discrete values, will give a positive result. All the other analysing functions with the "wrong" periodicity gives a 0 answer.
The idea is to use a computing machine to hit the observed set of values with as many analyzing functions as we like. The ones that give a non-zero answer is related in some ways to the original observed set of values.
This is the fundamental idea of discrete fourier transform. The rest of the fourier analysis is mainly about finding the bounds and limites, such as how high a frequency we need to test, what is the minimum resolution of data to get a clear positive answer, and what does the answer '4' really mean in the second calculation above, etc...
For a complex valued data, we just think of it as a 2-D array. So we repeat the same process as above two times, once for the set of real values, and another for the set of imaginary values. A complex number is just a 2D array [r_part, i_imag], and at the very end we can take the Pythagorean distance sqrt(x^2+y^2) of the real and imaginary components for each frequency bin to get the final answer.
Hope it helps.