For reference: here’s a better work on filibusters
The rule that allows “modern” filibusters appears to be rule XXII-the cloture rule. under this, the process seems to be that, once a filibuster is started, a motion can be made to hold a vote to end debate (for cloture)-which ends the filibuster, if approved. (it has to be made on the issue before the senate).
Once the motion is raised, however, the vote is held two days afterwards (“on the following calendar day but one”). However, in the intervening time, the senate doesn’t need to debate the question on which cloture is sought (i.e. the bill/nomination that’s filibustered). So they can get on with other business for two days (and the two-day period can, and often is changed by unanimous consent-so it could be weeks before the vote for cloture comes up).
Once the vote is taken, and (assuming it loses), the filibuster continues–but another motion for cloture can be raised. Rinse, repeat.