Why is fourier analysis used in doppler effect?
The fourier transform of a signal (like sound) will show you the sound in the frequency domain. In other words, the independent variable, or x axis, will be in terms of frequency (such as hertz or radians per second), and the y axis will show the magnitude of those frequencies.
One reason that the fourier transform is useful is that it will show what frequencies are present in a signal and in what quantities. When used in the experiment you described (if I understand it correctly), you could help demonstrate the doppler effect by showing that the sound source is outputting a tone of a constant pitch, and in the recording of the sound source moving in the circular pattern, the fourier transform of the recording will show that the frequency is not constant, and appears in significant quantities in the sound.
If you record the sound as the source moves in a circle, then the fourier transform of the sound should appear to be high in a range close to the pitch that the source is outputting.
A tool I frequently use for computing the fourier transform of an audio file is matlab. I am sure there are many tools online to help you compute this as well if you do not have access to matlab.