Stripping the FCC of being able to levy fines renders the department pointless.
If SCOTUS upholds, it won't remove the fines, it will just require them to take you to court, then get the court to levy the fines.
This isn't completely silly. They're basically pointing to the fact that various executive branch agencies have been doing legislative and judicial things, making rules and judging when/how those rules apply. Justice Kagan gave some interesting history about that in the recent Trump v Slaughter hearing. She made the point that we've made a kind of a deal with ourselves over much of the last century, allowing Congress to delegate legislative and judicial power to executive agencies, with a couple of caveats.
The first is that the agencies are restricted to making rules and adjudicating issues only within the narrow scope of authority granted by Congress (e.g. the FCC should only make and enforce rules about communications, specifically about the use of the airwaves). The second is that the agencies are somewhat independent, meaning the president has very limited ability to tell them what to do. These caveats maintain a separation of powers. It's not the same separation of powers that is described in the Constitution, because the Constitution separates powers by function (legislative vs judicial vs executive) whereas the "deal" separates powers by subject matter. So the FCC has been exercising legislative, judicial and executive powers, all three, but only within their scope. The FCC can't do anything about, say, fishing, or mining, or tax collection.
This is all arguably unconstitutional, especially with respect to judicial power, because Congress can't delegate what it doesn't have... but it works. It gives the agencies the powers they need to do their job effectively, and it also prevents any one person or organization from collecting all of the powers of all of the agencies in one place, because concentrating that much power in one place is dangerous, and specifically what the authors of the Constitution wanted to avoid. So you can quite legitimately say that it violates the letter of the Constitution, but is very much in line with the spirit.
In hindsight, it's really unfortunate that we never amended the Constitution to make it explicitly allow and codify such an alternative separation-of-powers strategy, but we didn't.
Now, we have a president who wants to exercise direct control over all of those disparate, mostly-independent agencies. This is why the argument about whether or not the president can fire the heads of independent agencies is a huge deal, because if he can then he can exercise direct control over all of them, gathering all of their disparate powers to regulate, well, everything, into one pair of hands.
That's what Slaughter is about; whether the president can directly control all of these quasi-independent agencies.
This is about another aspect of the same thing: Can the agencies adjudicate the rules they make, or do they need to go to court every time? If they win, this will actually undermine Trump's goal in Slaughter, because it will take power away from the agencies and give it to the judicial branch, power that he is trying to gather up for the presidency. Well, for himself.
It will also seriously undercut the ability of the agencies to do anything by making applying regulations vastly slower and more expensive. At least, that's what it will do when the targets are big corporations with plenty of money to fight those legal battles. When the targets are small companies or individuals without such deep pockets, it won't have much effect, except maybe to force those small targets to pay lawyers as well as fines.
Woohoo! PIRATE RADIO EVERYWHERE!
Sure, if you can afford protracted legal battles against the federal government. And if you win.