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I would say that the explanation for the servant's suffering is contained in the verse cited in the question (Isaiah 53:10), which indicates that it was meant by God to make a sufficient atonement (that is, to pay the full penalty) for the sins of all humanity from eternity past to eternity future. The willing endurance of this undeserved suffering by the servant (that is, Christ), in turn (and as also described in the same verse) caused God to take pleasure in Him and to prolong His days (through His resurrection), as well as enabling Christ Himself to see the multitudes who would subsequently be saved through faith in Him.. The author of Hebrews later spoke of the same willing endurance (as well as the motive for it) in Hebrews 12:2.
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