Why is the size of the feature bagging sample typically the square root of the total predictor set size? Applying feature sampling to each tree in the context of a Random Forest model, if the set of predictors is of size $p$, why does the size of the sample of predictors for each split typically be of size $m \approx \sqrt{p}$?
I get why the sample size should not be enough to allow to choose among the majority of the predictors, but why this precise value?

It's just a recommended default value. Leo Breiman observed that this value tends to work well on the classification problems that he worked on, but I'm not aware of any rigorous demonstration that this value must work best on all problems; indeed, the discovery that a different value works better on a specific problem seems to show that such a proof would be impossible. For RF regression, he suggests a different value. In short: you should tune $m$.
