Where do you get "If $\bar A A$ is assumed to be true" from?
The reasoning must be something like: Assume $\bar A A + \bar A B$, that is, at least one of $\bar A A$ and $\bar A B$ is true. But $\bar A A$ is never true, so it must be $\bar A B$ that is true. Then, in particular, $B$ is true.
This feels slightly circular, but that depends on exactly which logical axioms one already has (and it sounds vaguely like the book you quote from doesn't bother to present which logical axioms it starts from in the first place). However, if you already know "$\bar A A$ is always false", then this is a valid way of concluding $\bar A A \Rightarrow B$.
On the other hand, how do you know that $\bar A A$ is always false? If that is by truth tables, it would have been simpler to use truth tables to show that $\bar A A\Rightarrow B$ is a tautology.
On the other other hand, a "book of probability theory" is likely not intending to be a self-contained account of logic in general. I imagine that the history of the remark you quote must be that an early draft of the book just used the known fact that a contradiction implies everything, but that some readers in the test audience were confused because they didn't already know that. So the author inserted the briefest explanation that he had a reasonable expectation would satisfy those readers. A sceptical but non-sophisticated reader will probably agree that $\bar A A$ is always false, and will probably be accept reasoning about implication in the "assume such-and-such, then this-and-that" style. However, non-sophisticated readers have been known to balk and require extended explanations if presented with the actual truth table for $\Rightarrow$.