I understand it best by considering some of the ways that constitutional rights are limited.
First, the First Amendment is limited by its text. When it says that "Congress shall make no law..." it is limiting freedom of speech to freedom from interference by the federal government. So if you advocate genocide, the First Amendment stops the FBI from arresting you, but it doesn't care if Twitter suspends your account. The First Amendment is only a shield against the government.
Second, the First Amendment is limited by the rest of the Constitution. It says Congress can't infringe on freedom of speech, but the Sixth Amendment says that a criminal defendant can force witnesses to come into court and testify about what they know. So if I saw a crime, the First Amendment says that I don't have to talk about it, but the Sixth Amendment says I do. Who wins? Because the right to a fair trial is considered the most powerful right in the Constitution, the Sixth Amendment prevails and I have to talk about what I saw. So even against the government, the First Amendment is a shield against the government, but only when the government isn't using a sword that can pierce it.
And third, the First Amendment is limited by its purposes. The government can require or prohibit speech in certain circumstances even when there isn't a constitutional mandate justifying it. Congress can prohibit you from threatening the president, hiring a hit man, sexually harassing co-workers, and making deceptive sales pitches, even though all those things are typically done through speech. The First Amendment is a shield against government restrictions on speech itself, not against restrictions on the effects that our speech has or against restrictions on our speech that are only incidental to some other salutary purpose.
I understand Dworkin to be talking about the way these categories overlap and interact.
As it was written, "freedom of the press" meant a lot less than modern journalists like to think. It was just a ban on requiring presses from getting pre-approval for what they wrote. It still provides that same protection, and that protection is nearly absolute. You might find a local yokel judge to approve an injunction against printing a major expose of government misconduct, but that injunction is going to be reversed by a higher court in about eight minutes.
But when Dworkin wrote his book, the limits on "freedom of the press" were a really hot topic, and the courts were constantly picking up newspaper cases to consider what else it might include. Can reporters refuse to testify to a grand jury? Can they refuse to reveal confidential sources? Can they demand access to prisons to report on the conditions inside? Can they demand interviews with specific inmates? Are they entitled to watch a judge conduct jury selection in a criminal trial? Are they entitled to report on court proceedings despite a judge's order against reporting? Are they entitled to copies of government records? Can they defame people with impunity?
You can make a legitimate argument for answering "yes" to any of those questions. But every time you say yes, you're also shifting power to the press at the expense of someone else -- criminal defendants, law enforcement, private citizens, etc.
Dworkin is saying that the First Amendment can retain the most strength when it is narrow and clear. We already know what the rule is supposed to do and how it's supposed to work, so there's not a lot of room for negotiating over those other limitations we discussed above. You can try balancing the right to print against other constitutional protections, but if it's the government trying to stop publication of a story, the First Amendment is a shield that can almost never be penetrated.
But that's not true when we take that a step further to say, "The point of freedom of the press is to inform the public, and the press can more fully inform the public if we decide that freedom of the press also means that reporters can do X, Y and Z." Now we're getting away from the core promise of the First Amendment, and we're bringing in tougher questions about what the amendment can do.
Dworkin's concern is that moving in that direction creates a lot of new space for haggling over not only those new rights, but also over the original rights. If a reporter has a shield against a prosecutor's subpoena, it's not a very strong one because of all the interest balancing we have to do to figure out whether the shield even exists in the first place. And if we're doing that interest balancing to figure out how strong the First Amendment shield is in that context, why can't we start doing interest balancing on everything else?
Besides making people comfortable with interest balancing exercises that are bound to weaken the First Amendment's protections, letting the First Amendment do so much work that requires interest balancing in the first place means that you're going to generate some degree of resentment among people who think the interests should have been balanced in the other direction. This undermines public support for the First Amendment and makes it easier both for police, legislators, and the courts to ignore or work around it when adhering to it would be less palatable.
tl;dr: Shields that are misused can break. To keep your shield in the best condition, use it sparingly and only against the weapons that it can withstand.