How do electrical body fat detector work?
I have designed cardiac output meters, which is something related to what you are asking, although more complex. Cardiac output meters measure transthoracic electrical impedance, and --knowing that the blood is a good conductor of electricity-- they correlate conductivity with the amount of blood present in the volume between the sets of electrodes, at any given time. They provide as a result a waveform of the instantaneous flow of blood through your heart, in liters/minute. Body fat "meters" provide a rough estimate of the proportion of fat in your body. I'm not very familiar with the latter ones, but I know they also work by measuring electrical impedance.

To measure electrical impedance, you inject an AC current through some portion of the body of a patient. You use two electrodes to inject current, and two electrodes to measure voltage. The amplitude of the AC current is kept constant, and the amplitude of the sensed voltage is proportional to the instantaneous impedance. Typical frequencies used range from 20 kHz to 100 kHz. If the current has to go through your heart, a maximum of 400 \$\mu\$A\$_{rms}\$ is allowed, at 50 kHz. If only the magnitude of Z(t) is needed, best is to use a synchronous demodulator, to extract the amplitude information from the sensed voltage. If phase information is needed, you need more complex methods (like very fast ADCs and FFT).

There is no way that a circuit like this one can be done by an amateur, so forget about this. You need to sense a few microvolts (with a good instrumentation amplifier), and at the cables right next to the sensed voltage you have your excitation signal, spanning several volts. You need a very good schematic, and a better PCB design. I ended up with 4 or 5 ground areas, plus the same number of -5 V and +5 V areas. One of my toughest challenges, but the results were very good.

All this was for electrical bioimpedance in general, and for transthoracic impedance in particular. Knowing that fat is a relatively bad conductor of electricity, body fat estimators must correlate conductivity with amount of fat between the electrodes. I guess they use weight and volume data to help in that correlation, and data from MRI scans to have some absolute references.