Henrik Forstén has a nice build log on his newest version of this homemade 6 GHz FMCW radar:
Frequency Modulated Continuous Wave (FMCW) radar works by transmitting a chirp which frequency changes linearly with time. This chirp is then radiated with the antenna, reflected from the target and is received by the receiving antenna. On the reception side the received signal that was delayed and undelayed copy of the transmitted chirp are mixed (multiplied) together. The output of the mixer are two sine waves that have frequencies of sum and difference of the waveforms. The frequencies of the received signals are almost the same and the sum waveform has frequency of about two times of the original signal and is filtered out, but the difference waveform has frequency in kHz to few MHz range. The difference frequency is dependent on the delay of the received reflection signal making it possible to determine the delay of the reflected signal. The electromagnetic waves travel at speed of light which allows converting the delay to distance accurately. When there are several targets the output signal is sum of different frequencies and the distances to the targets can be recovered efficiently with Fourier transform.