“Yet it pleased Yahweh.” Israel (indeed, every human) sinned, and the only way for God to atone for that sin and grant everlasting life to people was for someone—a human without blemish (i.e., sin)—to die in place of the other humans, because the wages of sin is death (Rom. 6:23). Thus it pleased God to be able to have His Son die so that everyone else could live. As Romans 5:18 says, “So then just as one transgression [Adam’s sin] resulted in condemnation for all people, so also one act of righteousness [Christ’s atoning death] resulted in righteousness that brings life for all people.”
“If his soul makes” This is an idiom for “if he makes,” using the word “soul” to represent the person himself. The Hebrew verb form can be second person masculine, “you,” or third person feminine (“she” or “it” referring back to the word “soul” which is feminine, but which refers to the soul the Messiah). In this context it makes more sense that it is the servant (the Messiah) who is offering himself in obedience to God, and will be rewarded for it. So the idea is, “If his soul makes [itself] a guilt offering….”
“a guilt offering.” The Hebrew word is used of the “guilt offering” (some versions translate it “trespass offering”) in Leviticus 7 and 14, and in other places in the Old Testament. Jesus Christ was offered as a sacrifice that paid for our guilt and sin. This is also stated in the New Testament in 2 Corinthians 5:21. Israel sinned in so many ways that many of the sacrifices in the Mosaic Law applied to the death of Christ. Here in Isaiah 53:10, God specifically mentions the “guilt offering,” emphasizing the great deprivation of God and other humans when people live in sin. For example, God was deprived of much of Israel’s service to Him due to their sin, so a guilt offering was certainly appropriate.
“The guilt offering was a specialized kind of sin offering required in cases when someone had been denied his rightful due. …The offerer’s part in the ritual was probably identical to that of the sin offering…As with the sin offering, the animal went to the priest as food. …The guilt offering was commended in instances when another party had suffered some deprivation. [For example, in the case of a leper] the LORD was deprived of the service due from the infected person so long as his disease kept him outside the pale of the ritually clean society (Lev. 14:12-18)” (Zondervan Pictorial Encyclopedia of the Bible, “Sacrifices and Offerings”).
“and the pleasure of Yahweh will prosper in his hand.” The phrase is somewhat idiomatic, and it means that the good things that Yahweh desires will prosper and succeed “in the hand” (i.e., “under the authority and control”) of the Messiah.