Strictly speaking, nothing went on in the courtroom for this case. However, I can provide some context for the case and its outcome and explain the proceedings.
This case was decided in 1952, when the United States was in the grip of hysteria over Communism, hysteria fanned by Senator Joe McCarthy, who chaired a committee that went looking for Communists everywhere and that destroyed the lives of some very good people in doing so. During this period, many states and cities instituted regulations that prevented "subversive" people from holding public office or taking civil service positions. Teachers and others were expected to take oaths of loyalty to the United States, and anyone who belonged to a Socialist or Communist organization was barred from teaching.
Adler and others argued that this requirement, for teaching in New York City, which arose under a statute referred to as "the Feinberg law," was violative of the First Amendment, interfering with freedom of speech and assembly, depriving those subject to the requirement of due process, and also unconstitutionally vague with its reference to "subversive" organizations.
The reason nothing went on in the courtroom is that the plaintiffs asked for a declaratory judgment regarding the constitutionality of a civil service regulation for teachers, and the court found for the plaintiffs in a judgement on the pleadings. What this means is that there was no need for evidence to be presented in a trial, since the language of the regulation was all that was at issue, and the trial court was able to make a ruling on the complaint, answer, and briefs it had before it, without holding a regular trial. When there is no disagreement on facts, there is sometimes no need to have a trial, and this was such a situation.
The trial court held that the regulation was unconstitutional, and the city of New York appealed. The New York appellate court reversed and held for New York City, upholding the regulation, and the highest appellate court of New York upheld the regulation, as well. Because of the constitutional issues involved and because this was really an issue of concern across the entire country, the United States Supreme Court heard the case. They upheld the ruling of the New York court and found the regulation to be constitutionally valid.
It is not clear whether or not such a ruling would be made today. Since the area of free speech, particularly free political speech, has been expanded substantially since that era, and since the breakup of the Soviet Union and glasnost, we have not had any serious Communist witch hunts in the United States. However, as events unfold in Eastern Europe, with what appears to be actions by Russia that are strongly reminiscent of the Cold War era, the pendulum could easily swing back the other way, leading to rulings that are similar to this one.