Can someone help explain what this FFT workflow is doing to my signal, and why it works?
The answer from Bird has it right.

The described setup is an implementation of a lock-in amplifier.

Lock in amplifiers act as very narrow band filters for detecting the presence of a known signal that is buried in noise.

If you multiply two signals (both simple sine waves) you get a new signal that contains a frequency that is the sum of the original two signals and a frequency that is the difference of the original two signals.  (1000Hz * 200Hz gives an output of 800Hz mixed with 1200Hz.)  This process is used in radio receivers to bring the received radio signal down to a frequency where it is more convenient to do further processing.

Your LabView program makes use of this effect to detect the amplitude and phase of the signal in comparison to a reference signal.

The difference of the input signal and the reference signal is a signal of 0Hz - that's DC.  You get it out of bin 0 in your FFT results.

If the frequency of the input and the reference are not the same, then the difference lands in another bin and is ignored by your detector.

That also makes sense of removing the DC offset from the sample sets - any real DC in the input would cover generated DC from the heterodyning step.

As to why your device is doing that, it is probably so that it can recover a transmitted signal to tell how much of it is coming back to the receiver, and do it pretty much no matter how weak the received signal is or how much noise is present.  

Since you say it also takes the phase of the output, it may be using it as part of measuring a distance.

You don't say what the result is used for, or in what conditions you use it so there's not really much more I can say about the "why."