I think the theory works on the thought that many soaps are super-fatted, newer soaps do this now to introduce beneficial lipids, but I would assume that the older soap makers would do so because (at the time) soap companies wouldn't have surface chemists working for them and as such would err on the cautionary side to eliminate as much lye as possible in the solution to keep customers returning. Of course this was at a time when people actually had pride in their work and knew that a name meant something, but that's another rant for another day.
Of course the glycerine produced from the fats could also be what is helping the strop, I don't know. I'm not a leather person.