Synthetase is bringing up some good points. I have not been able to watch the video yet, but there are a variety of reasons you use these filters.
Time for me to start rambling on about stuff nobody cares about
This is VERY much my realm of expertise, being a bionanomaterials chemist with my recent focus having been on bioconjugation of fluorescent materials.
The first is that the stokes shift between the excitation and emission is usually around 10nm difference. (So a dye that emits at 545, would have a peak absorbance around 535nm). This means that you need a solid filter set to prevent the laser light from actually entering the detector and obscuring any of the signal you are trying to detect. For microscopy, we often use filter cubes which have a variety of filters to handle the various wavelengths needed. Here is an example of a common
filter cube for FITC The blue is the filter for the excitation wavelength, the green is the beam splitting filter (it reflects the excitation up into the sample, while letting the return fluorescence through) and the red is the emission filter.
Also, organic dyes have a phenomenon which is called a red tail. What this means is that their emisison profile is not symmetric. If you see the following image:
The solid black line is the emission spectrum, you can see how instead of returning to baseline it has this tailing off effect in the longer wavelenghts. This bleeds into any other channels that are to the red side of the dye. This is a VERY big issue in flow cytometry and microscopy, as well as many other polychromatic analysis methods. The following image shows the spill through:
You can see in particular how the green dye overlaps with the yellow dye, giving a significant signal represented in the yellow channel. This signal is represented by the green hatched triangle.
Compensation is damn important in flow cytometry, so much so, that my old place of work made LARGE amount of money selling compensation beads. Sub-um scale beads that helped aid in the compensation of the instrument. This shows before and after compensation of the instrument to show you how important compensation is. For these signals you want them entirely parallel with their axis, because any deviation from this is considered a positive signal.
As you can see, compensation can make a pretty big difference. Hopefully, my rambling helped explain the use of the filters in the instrument.