What you're really asking is about the history of the First Amendment. For our non-American friends:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances. bolding mine

The First Amendment does not protect all speech. In Schenk v. US (1919), Oliver Wendall Holmes famously wrote:

The most stringent protection of free speech would not protect a man in falsely shouting fire in a theatre and causing a panic. [...] The question in every case is whether the words used are used in such circumstances and are of such a nature as to create a clear and present danger that they will bring about the substantive evils that Congress has a right to prevent. bolding mine

This case was about convincing men not to join the army during a time of war, but the famous quote clearly points out that Congress has a right to prevent "substantive evils" and the First Amendment has been clarified in this light many times. For your question, you want to know about obscenity (fuck yeah!).

The Hicklin Test was the predominant test prior to 1957. The Hicklin Test came from an English case but it was based on common law that was in use in the US. The main idea was that any material tending "to deprave and corrupt those whose minds are open to such immoral influences" was obscene and, in the US, not protected by the Constitution. (Fun fact - the British case Hicklin was about publishing an anti-Catholic book that talked about dirty questions a priest might ask a woman during confession!)

The Hicklin test was replaced in 1957 with Roth v. United States. Samuel Roth had a company that sent out pornography erotic art and he was arrested (numerous times) for it. Another fellow, Dave Alberts in California, another fellow with prurient interests, was also arrested and convicted for similar crimes. The Supreme Court upheld their convictions but replaced the Hicklin test, which they deemed too broad, with a new test. Material whose "dominant theme taken as a whole appeals to the prurient interest" to the "average person, applying contemporary community standards" was now obscene. This was a narrower definition than the Hicklin test but it would only last a few years.

Jacobellis v. Ohio (1964) was about a movie theater owner who showed an obscene movie and it gave us one of the most famous quotes from Justice Potter Stewart:

I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description; and perhaps I could never succeed in intelligibly doing so. But I know it when I see it, and the motion picture involved in this case is not that.

The Court could came out with four different opinions on this case and the only thing that could really be agreed upon was that the First Amendment didn't protect "hard core pornography."

Memoirs v. Massachusetts (1966) added more confusion to the mix. It added to the Roth test by saying that obscene material had to be "patently offensive" and "utterly without redeeming social value."

It wasn't until Miller v. California (1974) that the Court came to a decision on what obscenity meant. The Miller Test asks three questions:

Whether the average person, applying contemporary community standards, would find that the work, taken as a whole, appeals to the prurient interest; Whether the work depicts or describes, in an offensive way, sexual conduct or excretory functions, specifically defined by applicable state law; and Whether the work, taken as a whole, lacks serious literary, artistic, political, or scientific value.

If these three criteria are met, it's obscene and the government can regulate it. It's important to note that the third point is critical - if the work contains any literary, artistic, political, or scientific value, it cannot be banned. If a person can show value, then the item is not obscene.

From Miller forward, the Supreme Court has only restricted more items. New York v. Ferber (1982) allowed states to ban sexual depictions of children even if it was not obscene. (The case involved a man selling videos of boys masturbating) Osborne v. Ohio (1990) allowed states to ban the possession of child pornography for most of the reasons it outlined in Ferber.

Back to your original question - how could people justify censoring music, movies, and television? They use the Roth and Miller tests as their basis. Do you think that Tipper Gore's "contemporary community standards" would include awesome songs such as Ice Cube's It Was A Good Day or NWA's Fuck the Police? Some of the media could easily be addressed by all three prongs of the Miller test - Striptease (1996) and Showgirls (1995) come to mind - while songs generally wouldn't, with a few exceptions. The big push was for Congress to pass laws against what these folks perceived to be obscene and they believed that the Court had allowed it. If Congress had passed laws and it ended up before the Supreme Court, it would not have been a slam dunk for the First Amendment. The Court's rulings mainly focused on sexual material. Does a song about killing cops have political value or is it an incitement to violence? The argument that the groups in the 80's and early 90's were making is that no, they don't have any redeeming value and they shouldn't be protected.

Here is a video of Frank Zappa testifying before Congress on the issue of censorship in music. He testified, along with John Denver and Dee Snyder. All of them spoke quite well (IMO) and the plans were scrapped and a voluntary system came about with the parental labels now on CD's to go along with MPAA ratings. In 1996, Congress called on television stations to come up with a voluntary system and in 1997 that system went into effect. So in the end, it was a wash - the government didn't pass any laws, the industries developed their own systems to warn people to "think of the children" when buying music or watching TV, and everyone was able to claim victory.