Here's the history as the Institute has it on its website
The key to understanding the development of the Frankfurt School is to understand some of the history of the social sciences. The Frankfurt School had as its mandate the scientific study of society and culture. Recall the earlier discussion on the nature of the social sciences. This now needs to be expanded.
The earlier discussion established that studying humans, either individually or in groups, could not proceed in the same way that the natural sciences proceeded. We might be able to explain the natural world, in the sense of finding causal connections, but we needed to understand the human world.
When the Frankfurt School was being built in the 1920s, the state of the social sciences was fairly rudimentary. That does not mean that people hadn't been asking about how to understand the social world - that had been true since the times of Plato. But as disciplines within the university, the social sciences were recent.