Another reason this subject is taught, besides the topics themselves, is that it develops your logical reasoning, abstraction, and visualization muscles. For example the first problem, he has to argue that L concatenated with L-reversed is regular, and he uses general properties of regular languages to do it. Then to argue the prefix and suffix parts, he has to utilize the automaton for L by making copies of it and drawing transitions between them. A final example is the famous and dreaded "pumping lemma" that gives practice understanding alternating quantifiers: there exists a such that for all b, ....