I'm trying to understand why most modern FT-IRs have three interferometers. Does this improve the signal in someway or does it have another effect?
I use FTIR in Physics, pretty much top-of the-range commercial instruments (actually more than just IR - ours go down to the UV and would do THz if we had the detectors; this isn't uncommon so I'll mainly refer to FT rather than FTIR). There's definitely only one interferometer in each of our systems (and the ones used by our friends in Chemistry). In astronomy FT spectroscopy has been used on the ground, but the best IR work these days is done from space using dispersive instruments (e.g. Spitzer's IRS).
The only common commercial use of three series-connected spectral devices (trying to coin a deliberately vague term) is dispersive instruments, which use three diffraction gratings for high resolution spectra and, for Raman spectroscopy, laser rejection with low stray light. A good benchtop FT instrument (with one interferometer) can give just as good resolution, though slowly and without the dynamic range. I mention this because if you were giving a lecture introducing a range of systems and techniques, and skipped a slide or two, you could jump from FTIR to triple dispersive systems and cause a lot of confusion.