As Rob has explained, an RGB working space (RGBWS) defines how images are separated into color and brightness components (speaking more rigorously, how chrominance/luminance separations are performed). In general, this is useful mainly during the nonlinear phase of post-processing.
The RGBWS implementation in PixInsight comes from the early days of PixInsight LE, back in 2002-2004. Here is a quite complete description:
https://pixinsight.com/doc/legacy/LE/14_color_spaces/color_spaces.htmlPlease be aware that I wrote the above document (as well as the entire LE documentation) more than 15 years ago. All has changed dramatically since then, including me and how I understand image processing and the PixInsight project, and basically everything inside and outside PixInsight
The idea of using two separate RGBWS, one for color management (acquired through color profiles and implemented by a color management system (CMS)) and a second one for luminance/chrominance separations performed during image processing tasks (implemented as a global RGBWS and the possibility of defining a different RGBWS for each image), made a lot of sense at the beginning of this century. However, I think it is questionable today, at least in the way it is currently implemented. Definitely, this part of PixInsight is subject to a strong critical analysis in my head.
Some people suggest setting the RGBWorkingSpace coefficients at equal values. Is this necessary? And if yes, when and why?
In most occasions it is not necessary. A custom RGBWS can be useful to perform nonstandard RGB-to-grayscale conversions, or to apply the CurvesTransformation and ColorSaturation tools to modify lightness, chrominance and color saturation components with more control. Read the document linked above, where you'll find some examples (very old ones, but still useful to describe the fundamental underlying concepts).