not a teacher
"Thus it was that in the 1930's the legal arm of the National Association for the Advancement of Colored People began plotting a litigation strategy to force the courts to confront head-on the evils of official racism."
I'll try to explain this, although a teacher may have to tidy up my response.
There are two separate terms here: "Thus it was…" and "that …".
Here, "that" is a conjunction.
"Thus it was…" implies that the information in the previous statement had a causal effect on what is about to be stated. In your example, the previous statement says that civil rights laws do not enforce themselves, they must be made to work by people in the everyday world taking action. The writer goes on to say that that's why the NAACP began their campaign.
To paraphrase: People must ensure the enforcement of civil rights in the everyday world. It was because of this that in the 1930s the NAACP began plotting a litigation strategy… etc.