This article, unless I'm missing something, doesn't actually explain why the algorithm works so well. It just gives the back story that led to the journal paper explaining it.
We introduce the smoothed analysis of algorithms, which is a hybrid of the worst-case and average-case analysis of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We measure this performance in terms of both the input size and the magnitude of the perturbations. We show that the simplex algorithm has polynomial smoothed complexity
https://arxiv.org/abs/cs/0111050
The two authors won the Godel prize in 2008 for this work.
I had completely forgotten I used to solve by hand at school until I looked it up: the article didn't jog that memory at all.