The idea has to do with the "nuclear option".
A filibuster is strong enough to stop a Supreme Court nomination, but that sets up a conflict. If the Democrats were to filibuster, the Republicans would have no choice under the current rules but to withdraw the nomination. However, that doesn't mean the Republicans have no choice. They can change the rules.
The filibuster isn't a constitutional thing. It's not a law. It's just a Senate rule. It's an agreement that has been in place for about 200 years. It sort of says that if you really, really, care about something, you can force a 60 vote majority for something. Well, it's not that simple, really, but that's basically it. For 200 years, it worked and served a purpose, specifically because it was used sparingly.
Recently, however, it has been used not so sparingly. Now, it has been used, by both parties, to block anything they don't like. If the Dems try to use it hear, the Republican majority might just say we don't need filibusters anymore, and just eliminate them from the rules. It doesn't require the president to do anything. It doesn't require the House, or the courts. All it requires is 51 senators to say that the filibuster is no longer a useful thing. That would mean no more filibusters, ever. That's the "nuclear option". You get what you want on the upcoming important vote, but if you ever find yourself in the minority, and you will, you give up the ability to block things later.