Unam Sanctam is a short document carrying a long argument. Boniface VIII begins from the unity of the church, one body with one head, and reasons from that unity to a single governing authority. He takes up the image of the two swords, drawn from a passage in Luke and read allegorically since Bernard of Clairvaux, and gives it the hardest available reading: both swords, the spiritual and the temporal, belong to the church, and the temporal sword is wielded for the church by kings and soldiers at the direction of the priest. Where the two conflict, the spiritual power judges the temporal, and the spiritual power itself can be judged by God alone.
The closing sentence is the one that mattered afterwards. It declares, defines and pronounces that it is altogether necessary for salvation for every human creature to be subject to the Roman Pontiff. Later Catholic theology treats that clause as the definition and the rest of the bull as the argument leading to it, which is why the document still appears in doctrinal collections while its political claim is read as a product of its moment.
Almost nothing in the reasoning was new. Canonists had been assembling the case for papal superiority over secular rulers for two centuries, and most of the phrases Boniface uses are quotations from that literature. What was new was the decision to state the case at full strength, in a formal bull, in the middle of a quarrel with the strongest king in Europe, at the point when the papacy could no longer make the claim good.