The procedure which you quoted can be used to invoke the nuclear option and defeat the filibuster, but it allows for a Senator or group of Senators to engage in old school talking filibusters as a motion to appeal the decision of the chair is debatable in general.
The general rule in the Senate is that each Senator can be recognized for an unlimited amount of time twice on each issue. Suppose for the sake of this question a bill a majority wants to pass (say they consist of 50 Senators and the Vice-President). Typically to end debate the Senate invokes cloture by three-fifths vote of the total number of Senators Chosen and Sworn (60 currently).
There are two options by which a filibuster can be defeated through a majority vote. The first option is to have the presiding officer rule certain debates arbitrarily out of order, ie. say that each Senator is limited to speaking for 10 minutes on an issue (this would most likely be done against advice of the Senate Parliamentarian), but it still is the prerogative of the Chair of the Senate to act in this fashion. The Chair makes the initial ruling on procedure. This puts in place an interpretation that ends arbitrary long debate immediately. The minority in this case can appeal the decision of the chair to restore the old interpretation of the rules (allowing unlimited debate). Since, originally, the motion which was being filibustered was debatable, this appeal is debatable and thus the filibuster can be normally sustained because the Senate would never actually resolve this issue and the underlying bill would never come to a vote. However, the majority has at their disposal a motion to table the appeal. Tabling takes precedence and is non-debatable. Thus the majority, 50 Senators+Vice-President, vote to table the appeal. Thus by tabling the appeal the original ruling stands and Senators are now limited to 10 minutes debate on an issue.
The less destructive method of defeating the filibuster using the nuclear option is to follow the standard procedures to invoke cloture (file a petition as by the rules) and take the cloture vote. Obviously with only 50 Senators + Vice-President the majority does not fulfill the requirement of three-fifths of the Senators Chosen and Sworn voting in the affirmative. The majority appeals the decision of the chair announcing the vote to invoke cloture failed (this might actually need to be done before the actual vote is taken by asking for parliamentary inquiry of the required vote total and moving to appeal the decision of the chair then—would need to really dig down in precedents of when voting threshold rules must be appealed), but regardless the majority then appeals that three-fifths vote is required and says it should be majority instead. Since cloture is a non-debatable motion, the appeal is non-debatable. Then the Senate votes to change the threshold of cloture by majority vote and changes the threshold to be majority vote by overturning the decision of the chair and instead ruling that cloture has been invoked. This is a least changes type of change because unlimited debate still exists in principle, but now a simple majority can invoke cloture and end debate.
The passage you quoted assumes that either by collegiality the minority will allow the appeal to come to a vote or the appeal is done of a non-debatable such as cloture as I described above.
