You were missing that these texts talk about "worst case expected run time", not "worst case runtime".
They are discussing a Quicksort implementation that involves a random element. Normally you have a deterministic algorithm, that is an algorithm which for a given input will always produce the exact same steps. To determine the "worst case runtime", you examine all possible inputs, and pick the one that produces the worst runtime.
But here we have a random factor. Given some input, the algorithm will not always do the same steps because some randomness is involved. Instead of having a runtime for each fixed input, we have an "expected runtime" - we check each possible value of the random decisions and their probability, and the "expected runtime" is the weighted average of the runtime for each combination of random decisions, but still for a fixed input.
So we calculate the "expected runtime" for each possible input, and to get the "worst case expected runtime", we find the one possible input where the expected runtime is worst. And apparently they showed that the worst case for the "expected runtime" is just O (n log n). I wouldn't be surprised if just picking the first pivot at random would change the worst case expected runtime to o (n^2) (little o instead of Big O), because only a few out of n pivots will lead to worst case behaviour.