The English Wiktionary example for the sentence 'унести' says
Ве́тром унесло́ мою́ ке́пку.
It is translated into "The wind carried my cap away.". Why does it begin with the instrumental case? I'm sure that « Ветер унес мою кепку » would also have been acceptable. Why begin a sentence with this case? What is the rule for this?
This kind of passinve sentences goes like this
Object in Accusative + Verb in neutral gender + Subject in Instrumental
Examples, mostly from news and forums:
It is mostly for accidents and incidents. You cannot use these when the subject is animate. For example, you cannot say "Меня ужалило пчелой" because пчела is animate. You cannot say "Пушкина убило пулей на дуэли" because d'Anthès shot Pushkin on purpose rather than accidentaly. You cannot say "Его вылечило от туберкулёза горным воздухом" because it is "good" and also because it lasts for long time.
Words often exchange their order. For example, one can say both "кепку унесло ветром" and "ветром унесло кепку".
Passive sentences of this kind can be rewritten with passive participle, or as non-passive sentences:
While in the examples above the verbs are in past tense, they also can be in present or future :
I'm not a native speaker, but here's my understanding of it:
It's an impersonal sentence, i.e., there is no noun or first-person, second-person, or third-person pronoun in the nominative case.
Consider the sentence:
In the same vein you could ask, why does this sentence begin with the dative case. Except you probably wouldn't ask because this kind of sentence is already very familiar. Also, it doesn't necessarily begin with a dative; you could write for example "Иногда просто хочется покоя".
Sometimes Russian uses an impersonal sentence for some verbs where English uses a sentence with a subject, for example:
So instead of impersonal sentences that involve a dative which are very familiar from Russian 101, there can also be sentences that involve an instrumental (like your example), or an accusative (like the above example). The verb is in the third person, and if it's in the past tense then it's neuter.
Finally, consider the opposite situations, where Russian uses a straightforward subject+verb sentence but English has an indirect formulation with an artificial subject. For example:
Actually there's not that much of a mystery here. In fact, even in English (I'm saying "even" just because modern English has no cases at all) there are some hints that could help you.
First, let's consider the famous title "Gone with the wind". This is translated to Russian literally as "Унесённые ветром" not as "Ветер унёс". So when one say in English "with something" or "by something" it most likely expressed in Russian via instrumental.
Compare another famous English phrase "baptizing by fire" to it Russian equivalent - "крещение [чем? - instrumental case] огнём".
Now let's take a look at "gone" - what form of verb is this? This is a participle, or, in Russian, причастия - or, to be more precise, those are passive participle.
Here's a quote from wikipedia which nicely explains what it is:
Passive voice is a grammatical voice common in many languages. In a clause with passive voice, the grammatical subject expresses the theme or patient of the main verb – that is, the person or thing that undergoes the action or has its state changed...For example, in the passive sentence "The tree was pulled down", the subject (the tree) denotes the patient rather than the agent of the action. In contrast, the sentences "Someone pulled down the tree" and "The tree is down" are active sentences.
I'd be glad to say that if you understand this one thing you should remember is the usage instrumental case when needed but unfortunately this is not the hardest part. The hardest part is to memorize all this verb-derived forms that can be use in passive voice. But I guess this could be a part of a different answer for a different question which you'll definitely will have many if you'll keep on studying Russian ;)
UPD: As @YellowSky has mentioned, indeed, things are a bit more complicated since, strictly speaking "Унесло" here is impersonal rather than passive, which is slightly a different concept but I don't won't go too deep into that. It is not oversimplification to say that those two concepts are very closely related.
