给X is the Chinese version of accusative or dative case. 给X indicates X is passive, not active. Before or after a verb, it just shifts the action away from X. Otherwise, X would somehow also seem active and you may end up with a nonsense sentence.
What happens if you don't use it?
给他打电话 give him a call
打给他电话 give him a call
他打电话 he calls
给他送 send him/give him
他送 he sends/gives
她试图给他剪指甲，他不让。 She tried to cut his fingernails but he didn't let her.
(In German that would be: Sie wollte ihm die Fingernägel schneiden, doch er ließ sie nicht. 'ihm' is the dative case of German 'er' = 'he', corresponding to Old English 'him' and Modern English 'him')
她试图他剪指甲，他不让。 *She tried he/him cut his fingernails, ....
证明给我看 We could translate: Prove it to me. German: Beweis es mir. 'mir' is the dative case of 'me' in German.
But I think this really says: 证明给我看 Proof let me see. / Give me proof to see. / Let me see (some) proof.
As Modern English has lost most of the case system that Old English had, it bypasses that using a prepositional phrase sometimes. I wouldn't just generalize and say 给X is 'to X' or 'for X'. That won't work in many cases.