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.