Against Fascist dictatorships
This was written during WWII, as Europe was fighting to defeat the Nazi regime that had spread its reign across continental Europe. The minds of the authors were on "National Socialism" in Germany, and other forms of Fascism in general. They disagree with the premise of Fascist regime on purely ethical reasons. Because the health of a society is determined by how effectively the narrative of that society's culture helps the specific people in that culture to attain their personal enlightenment, then it would follow that the Fascist seizure of public narrative (through propaganda and strict laws prohibiting free speech), Fascism would be antithetical to the purposes of human progress.
Against Capitalist greed systems
Just because Adorno and Horkheimer oppose Fascism, that doesn't mean that they are friends with Capitalism, especially not that pernicious kind of Western Capitalism that is designed to only serve the interests of big businesses and pre-existing wealth. Those cultures are also inherently opposed to public enlightenment, Adorno notes, because just like the Fascists don't want heroes overthrowing their unjust system, neither do the Capitalists want heroes, because the upper class is exploiting the lower class. That's why the book has "Dialectic" in the title—as a reference to the same observations by Hegel that led Marx to invent Communism.
Against Logical Positivism
Adorno and Horkheimer spend the middle portion of these essays to dismiss the school of Logical Positivism, citing that they too had rejected the human need for narrative by consistently seeking to undermine the assumptions that allow humans to encounter narratives in a meaningful way. By the way, Logical Positivism was the school of philosophy rising to power in the universities in Europe in the mid-20th century. They also include Wittgenstein in their criticism, because his strange theories on language and human thought are also undermining meaning.
