The name Jesus is the Greek form of the Hebrew name Yeshua or Joshua and is associated with the confession that the LORD saves. A Jewish reader would naturally hear the child's name against Israel's long history of looking to God for deliverance.
The striking feature of the angel's explanation is its subject: Jesus himself will save his people from their sins. The promised salvation is deeper than political rescue; it concerns the fundamental covenant problem of sin.
Psalm 130:7-8 calls Israel to hope in the LORD because with him is plentiful redemption, and declares that he will redeem Israel from all its iniquities. Isaiah likewise repeatedly presents the LORD as Israel's Savior and Redeemer.
Matthew's wording therefore places Jesus within a divine saving pattern. The point is not merely that his name contains a reference to God, but that the saving action associated with God is said to be performed by Jesus.
Matthew begins explaining Jesus' identity before Jesus speaks a word. The angel commands Joseph to give Mary's son the name Jesus and immediately explains the name by the child's mission: he will save his people from their sins.
A prophet may announce God's forgiveness and a priest may administer the covenant rites associated with forgiveness, but Matthew's wording makes Jesus the acting Savior. The following Emmanuel citation in verses 22-23 strengthens the connection between God's promised presence and the saving work of Jesus.
The verse does not contain a direct statement that Jesus is God, so it should not be treated as an isolated proof text. Its importance is cumulative and contextual.
Matthew assigns to Jesus the saving work that Israel's Scriptures characteristically attribute to the LORD. Read with the immediately following declaration of Emmanuel, it is an early and strong indication that God's saving presence has come personally in Jesus.
Joseph obeys the angelic command and names the child Jesus.