I interpret this sentence's meaning as something like, "The letter ended up being seen by someone else."
From what I understand, the verb 見る is transitive, "to see." The verb 見られる should be its passive form then, "to be seen," which should be intransitive. The particle を indicates that the preceding noun is the direct object of a verb that follows it, right? So then, wouldn't 「他の人に手紙が見られてしまいました」 make more sense?
I have a feeling this might have more to do with the verb than the particle. Does 見る not lose its transitiveness when it is put in its passive form? If so, is を able to be used without a transitive verb then?
Where did I go wrong?