First, know that (as an Englishman) the author uses Liberal in the English sense, which is to neither be a defender of inherited privilege nor a proletarian revolutionary. A liberal is one who believes in meritocracy, in individual political and economic freedom, and in creating improved policies based on reason. He also deliberately puns “liberal education” in the traditional sense of deep study of history and classic literature. He believes that these two meanings of liberal are compatible, along the line of traditional arguments that liberal education “improves the mind”. However he spends much of his time chronicling changes in higher education during the last 200 years.
This takes us from a time where in the US:
As recently as 1900 only about 10% of Americans attended high school or a prep school. This rapidly changed as public high schools were opened. The huge growth in higher education began in the 60's.
The “anxieties” in the title refers to his discussion of what we should reasonably be worried about. I think he has good points about many criticisms of contemporary education being ill-founded. In particular, the sense that the education system is somehow performing worse than it used to is completely lacking in the historical perspective that such an argument would need. It may well be true that at a time when education was confined to cultural elites it was easier to maintain high standards and unashamedly inculcate the values of those cultural elites, but the scope of the educational system has been entirely redefined in the past century.