So many times I used to read this and put it to the back of my mind not wanting to invest any energy in understanding it - probably because I didn't need to. And in many ways, I still don't need to but explanations sometimes are useful.
These days, I don't often shoot exclusively strobe lit. In the studio, I blend ambient and strobe light. A kind of unpredictable concoction but there is no other way to get some of the effects that I want other than using natural light alone and carefully shaping it. The big disadvantage of that is that it's dependent on a certain amount of natural light. What if it's cloudy, what about the difference between 10am in summer and 10am in winter... Direct studio strobe can look good of course, in the right situation, but it has an artificial look that I sometimes don't get. Solution: blend natural with strobe, or rather supplement natural with strobe.
If you're using purely strobe at small apertures and high powers then your shutter speed settings won't matter at all as the ambient light in the room is usually too dim - i.e. strobe overpowers natural. And, most strobe durations are less than 1/1000 second duration so if the shutter's open for 1/250 of a second, who cares - exposure isn't going to be determined by that. It will be determined by aperture alone - rather aperture and strobe power output (oh, and distance of strobe to subject and ISO setting).
But... if you are using a very low strobe power, like the absolute lowest setting, and the room has a nice amount of ambient light (say ISO 400, 1/100s, f2.8) then your shutter speed WILL affect the result. I don't like shooting portraits at anything smaller than f2.8, for reasons of my own beyond shallow DOF. So supplementing that small but sweet amount of natural light with a strobe can create a situation that you can use no matter the level of natural light. How you actually use the strobe though is the important thing. Do you bounce it into the shadows, direct it at the shadow side, use softbox, umbrella, bare bulb - or do you just use the modelling light? I have my own recipes for all this stuff having experimented until the cows came home. All questions for another article...