The short version of what’s going on: you can prevent a vote from occurring in the Senate by filibustering it. It requires 60 votes to end a filibuster. Senator Frist is threatening to change the Senate rules in order to require only 50 votes to end a filibuster. However, changing Senate rules has always taken a 2/3rds majority vote. How’s Frist gonna get around that?
Well, he’s going to raise a point of order arguing that the filibuster is unconstitutional because it prevents the Senate’s Constitutional duty to advise and consent on judicial nominations. That point of order will go to the Senate’s presiding officer, who will be Cheney. Cheney will then say “You’re right,” and the vote to end the filibuster will take place.
However, Senate precedent says that Cheney can’t make the decision on constitutionality; rather, it should go to a Senate vote, which is itself subject to filibuster. And of course the Democrats would filibuster it. So Cheney has to break precedent and make a ruling. It’s fair to note that this does not break Senate rules, but Senate precedent is not unimportant either.
If you want the detailed look at this, start here and go on to this, this, this, this, this, and this. There are more posts in that series, but those are the ones that address Constitutional and Senate rules issues rather than arguing about the value of the filibuster itself and the meaning of “advise and consent.” Which are interesting questions, but not as relevant to this post.