The rarest tense in Italian
The honest answer first: in everyday life you will hardly ever need this tense. Almost nobody says it on the phone or writes it in a message. It lives in novels, in elevated prose and in historical writing. The goal here is therefore receptive: recognise it, place it, keep reading.
In real texts you find almost only the third persons: ebbe, ebbero, fu, furono. The form mostly occurs in stories about other people.
The context in which it appears
As a rule the trapassato remoto does not stand alone but in a subordinate clause with a time conjunction, and the main clause is in the passato remoto. It names an action that immediately precedes the next one. That is why appena fits so well.
The triggering conjunctions
Subordinate clause and main clause
Dopo che ebbe letto la lettera, la bruciò.
After he had read the letter, he burned it.
Non appena furono arrivati, cominciò a piovere.
As soon as they had arrived, it began to rain.
Quando ebbero mangiato, si misero in cammino.
When they had eaten, they set off.
Appena ebbi chiuso la porta, mi accorsi di aver dimenticato le chiavi.
No sooner had I closed the door than I realised I had forgotten the keys.
What people read and say instead
Modern authors often avoid the form, and in conversation the trapassato prossimo is the normal solution. For the sentence Dopo che ebbe finito, uscì you will therefore also find these variants.
The same statement, three variants
Dopo che aveva finito, uscì.
After he had finished, he went out.
Dopo aver finito, uscì.
After finishing, he went out.
Finito il lavoro, uscì.
The work done, he went out.
