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Rato means moment, while, or any short period of time in English and comes from the Latin word, raptus. Now raptus is a past participle of rapio which translates as "to snatch away or carry off." How did something so unrelated to time end up becoming the word for moment in English?
In short, what has "to snatch" got to do with moment?
It seems that beyond the "to snatch" concept (tirón, arranque) it had a figurative meaning of "an instant". Maybe because the original word meant a sudden and quick grasp, a violent action (as the words tirón and arranque suggest).
It seems that from there it came to be used as "a period of time a little bit larger that an instant".
-Rato, del latín 'Raptus': en su origen significaba ‘tirón, arranque’ y, figuradamente, ‘instante’, acepción esta última que viajó a América, al mismo tiempo que en España tomó el sentido de "un espacio de tiempo algo mayor que el de un instante"
I found this explanation (in Spanish) here (actually it also explains the origin of broma, in relation to this other question and other words whose original meaning has nothing or not much to do with the actual meaning. I think you might find that reference interesting, given your latest questions about etymologies. If you need further clarifications with the content of that site just tell me and I will edit the answer to enhance the explanation).
This is just my thought without a lot to back it up. I think it comes from amount of time that it takes to snatch something. For example, a purse snatching is something that happens very quickly, in a moment.
It would be similar to the phrase, "in the blink of an eye" that is used to describe something of a short duration.
