The Supremacy Clause settles which law wins when laws conflict, and it explains why state courts must follow the U.S. Supreme Court on questions of federal law.
The Constitution, federal laws made under it, and U.S. treaties are 'the supreme Law of the Land.' Judges in every state are bound by them, anything in state law to the contrary notwithstanding. When valid federal law conflicts with state law, federal law prevails.
Because federal law is supreme, the U.S. Supreme Court has the last word on what it means, and that word binds state courts too. The Court confirmed it can review state-court judgments on federal questions in Martin v. Hunter's Lessee (1816), and reaffirmed that state officials are bound by its constitutional rulings in Cooper v. Aaron (1958).
Supremacy only reaches federal questions. On the meaning of a state's own statutes and constitution, that state's highest court is the final authority, the U.S. Supreme Court does not second-guess it. This division is what makes the floor-and-ceiling idea possible.
Supremacy is about hierarchy between systems (federal over state on federal questions). Persuasive authority is about courts that sit outside your line of appeal, a sister circuit, another state, whose reasoning you may borrow but need not obey.
