The verb «подать» for a single action (as a form of «подавать» for repeated actions) means “to hand in”, “to give” or “to pass”, of which “to pass” seems to me the closest equivalent in meaning. Compare: “Could you pass me the salt?” – «Можете (не могли бы Вы) подать мне соль?»
Since we can “hand in / give / pass” smth through somebody else, or by post, or in any other way from rather a distant place, the idiom includes the adverbial modifier of manner «рукой» (“with one’s hand / by hand”) to show the maximum proximity.
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