Back in 2006, there was much talk about the so-called "nuclear option" over judicial appointments. As such, there was also some acedemic legal research done on the topic. One of those researchers was a guy named Lawrence Solum, who co-authored a paper titled Judicial Selection, Appointments Gridlock, and the Nuclear Option , that appeared in the Journal of Contemporary Legal issues. It may be found here.
In it, they detail how the nuclear option would work. And, it doesn't involve the 67 votes normally needed to amend the Senate Rules. In short, it's a back door parliamentary trick. You can read the whole thing if you like, but the relevant passage from their article is...
"The term “nuclear option” is not well defined, but it usually refers to the use of procedure—in particular, a ruling of the chair sustained by a simple majority—to achieve cloture without the sixty or sixty-six vote supermajority specified by Rule XXII. Though the nuclear option may be executed in a number of ways...
[t]he underlying strategy is that a Republican senator would raise a point of order that the consideration of judicial nominees may not be filibustered, and the chair—most likely Vice President Cheney, in his capacity as President of the Senate—would sustain the point of order. A simple majority vote would then suffice to win any appeal of the chair’s ruling, or to table any objections to the ruling
That's it in a nutshell.
The "nuclear option" entailed making a finding when it's unclear that the filibuster rule would apply. While the same approach could be used in cases where a filibuster would clearly be legitimate, the Democrats could also get their way by simply chasing all the Republicans out of the Senate at gunpoint before holding a vote. Neither approach would be any more legitimate than the other.