Learn Hardcore Dutch: Anna zet haar koptelefoon op, zodat het geluid niemand stoort. - Anna puts on her headphones so the sound doesn’t bother anyone.

In Dutch, opzetten is a separable verb. In a main clause you place the prefix (op) at the very end and keep the stem (zet) before the object. So Anna zet haar koptelefoon op literally splits opzetten into zet … op, meaning “Anna puts on her headphones.”

What does zodat do, and why is stoort at the end of the clause zodat het geluid niemand stoort?

Zodat introduces a subordinate clause expressing purpose or result (“so that…”). In such clauses Dutch follows the verb-final rule: the finite verb moves to the very end. That’s why stoort (“disturbs”) appears last in zodat het geluid niemand stoort.

Can I use om … te instead of zodat to express purpose? How would that sentence look?

Yes. For a purpose clause you can also use om + te-infinitive. The sentence becomes:
Anna zet haar koptelefoon op om niemand te storen.
Here om niemand te storen means “so as not to disturb anyone.” Generally, om … te highlights the intention, while zodat highlights the result.

Why does the sentence say haar koptelefoon? Could I say de koptelefoon instead?

Haar is the possessive pronoun “her,” so haar koptelefoon specifies that the headphones belong to Anna. If ownership isn’t important or has been established already, you can say:
Anna zet de koptelefoon op.
That simply means “Anna puts on the headphones,” assuming the listener knows which ones.

How could I express the same idea in the passive voice?

Use worden + past participle and introduce the agent with door:
Anna zet haar koptelefoon op, zodat niemand door het geluid wordt gestoord.
This means “Anna puts on her headphones so that no one is disturbed by the sound.” The passive shifts the focus onto the people being disturbed rather than on the sound.