My experience is in Lightroom, not Darktable, but LUTs just take an RGB value at each pixel and map it to a different RGB value. The LUT (look up table) is just a big table containing these mappings. They are most commonly used near the end of the development pipeline to do color grading. It's a common way to do something like emulate the color response of a specific type of film, even if you shot digitally.

So, typically I would do my developing (lens profile, crop, exposure, etc.) first, then apply a LUT. After the LUT, then I would tweak the color sliders if there are any I still want to adjust. The color is usually the last step of the process for me.

All that being said, these days I don't really use LUTs any longer. I am mostly doing product photography using a flash in a controlled studio environment. So, I pretty much have some develop presets all dialed in.