Learn Dutch: Aangemoedigd door de regisseur, ging de figurant improviseren. - Encouraged by the director, the extra started to improvise.

This is a past participial clause (beknopte bijzin). It acts like a shortened passive sentence. If we wrote it out fully, it would be something like Omdat hij werd aangemoedigd door de regisseur (Because he was encouraged by the director). By just using the past participle Aangemoedigd, the sentence becomes more concise. The unwritten subject of this action must match the subject of the main clause, which is de figurant.

Why does the main clause use inverted word order (ging de figurant instead of de figurant ging)?

The entire participial clause before the comma counts as the first position in the sentence. Because Dutch strictly follows the verb-second rule in main clauses, the main verb ging must immediately follow the comma. This forces the subject de figurant to move after the verb.

Why is ging translated as "started to" instead of "went"?

When the verb gaan is followed directly by an infinitive like improviseren, it often indicates the beginning of an action rather than physical movement. So ging improviseren means the extra "started to improvise" or "proceeded to improvise."

Does door always mean "by" in this context?

Yes, in passive sentences and shortened passive clauses like this one, door is the standard preposition to indicate the agent (the person who actually performed the action). Here, it shows that the director was the one who did the encouraging.