We will learn about the function and operation of the individual components that make up the direction finder.  However, before we do that, our education would be incomplete without a good understanding of the RF theory.  (this is radio, after all)
Phase Error - all those wiggly lines...
  A signal traveling from a transmitter will arrive at two different antennas at different times if the antennas are at different distances from the transmitter.   In the case of our direction finder, a worst case situation would result in the signal reaching, say, the left antenna, twenty three inches, or a little over a quarter wavelength, sooner than the right.  The signals would continue down the coax to the electronics.  In the case of the left antenna, the signal would have traveled X distance to the left antenna then about twelve inches down the coax to the electronics.  For the right antenna, however, the signal would have traveled X+another 23 inches and then twelve inches in the opposite direction down it's coax.  Now, once inside the coax, the signal cares not what direction it is traveling but will continue at roughly the same speed.  The signals  would emerge from the ends of the coax a little over a quarter of a wavelength out of phase as in figure 1.
If the difference in distance from the antennas and the transmitter is reduced, the phase difference of the received signal will also be reduced as in figure 2.
Frequency Modulation - fine magic
                (review time here)
As you recall, with FM we modulate

the signal by changing the frequency slightly around the main or carrier frequency.  The rate at which we change the frequency determines the audio frequency.  The amount that we change the frequency produces the amplitude.  Remember this, you will see it later.
                   (end of review)
  Suppose we have two signals arriving at the end of our two cables as in the top

  I would like to add here, that the waveform in figure 3 is not entirely accurate.  I have shown the antenna selects as occuring just two wavelengths apart.  In actual practice, if the carrier frequency were 146.00 Mhz. and the audio tone were 1 Khz., the carrier  waveform would be altered or modulated only once every 146,000 cycles.  Doesn't seem like much, but the radio seems to like it.  Amazing, fine magic.
A time machine....
  "But how does the tone get louder or quieter?" you ask.
  Refer again to figure 1.  You see that the L and R waveforms are separated by a  relatively large amount of time as compared to the waveforms in figure 2.  In the case of figure 1, when switching back and forth between the two waveforms, we will add or remove a lot of time from one cycle, changing the carrier frequency by a large amount, producing a louder tone.  (In fact, our D-F  rig is capable of exceeding the bandwidth of some receivers,  no big deal, but it produces a distorted audio tone.  This means the transmitter is somewhere way off to your left or right.)

figure 2

waveform of figure 3.  Now let us select between the signals at a 1000 Hz rate.  If we alternately chop hunks of each signal off and feed them to the reciever, the reciever will see a waveform like A&B in figure 3.  The switching produces little glitches like those circled but these are largely ignored by the reciever.  However, of greater intrest is the effect of the overall waveform.  Notice that in selecting antenna B we have added more time to one cycle of the combined waveform.  Conversely, in switching back to antenna A we cut some time out of one cycle of the combined waveform.  Now if we add time to a wave we reduce it's frequency, or if we take away some time from a wave we increase it's frequency.
The formula is:
       frequency = 1/time
where time is the duration of one cycle.
  The net effect of adding or subtracting bits of time by switching between out of phase signals is to introduce modulation.
  In our case, as stated earlier, our switching rate is around a thousand hertz.  This is the tone we hear superimposed on the audio of the recieved signal.

figure 3

figure 1