Learn Italian: Nessuno osa rompere il giuramento senza di te. - Nobody dares to break the oath without you.

This is a specific rule for a group of prepositions including senza (without), dopo (after), contro (against), and sopra/sotto (above/below). Whenever these are followed directly by a personal pronoun (like me, te, or noi), you must insert the preposition di. Therefore, "without you" becomes senza di te, rather than just senza te.

Why isn't there a non in this sentence? Doesn't nessuno require a double negative?

You only need the double negative (adding non before the verb) if the negative pronoun comes after the verb. Because nessuno is the subject and sits before the verb osa, the non is dropped. If you changed the word order to put the subject at the end, you would need it: Non osa rompere il giuramento nessuno.

Why is rompere attached directly to osa without a linking preposition like a or di?

The verb osare (to dare) takes a direct infinitive. This means it connects straight to the next verb without needing any preposition in between, much like how modal verbs (potere, volere, dovere) behave. You "dare to break" (osa rompere), not osa a rompere.