The Warren Court did these things through a long series of decisions in which the Court ruled in ways that radically changed American ideas of civil liberties.
During the Warren Court, there were many Supreme Court cases that are famous and important today. Many of these, at least arguably, increased personal freedoms. There was the case of Gideon v. Wainwright, in which the Court announced that indigent defendants had the right to an attorney. There was Miranda v. Arizona, in which the Court held that police had to inform people of their rights to remain silent and to an attorney when they were arrested. There was the case of Griswold v. Connecticut in which the Court found that the Constitution protects a generalized right to privacy. Most famously, there was the Brown v. Board of Education decision banning segregated schools. All of these can be seen as expansions of individual liberty.
There were also a few cases that expanded participation in the political process. Perhaps the most important of these was the case of Reynolds v. Sims, in which the Court required states to adhere to the idea of “one person one vote” in apportioning their legislatures.
In these cases and many others, the Warren Court handed down decisions that expanded American liberties.