The Gospel, Matthew tells us is at least in part that Jesus came to save his people from their sins. In most Evangelical circles this is the case and it is interpreted as a legal or juridical saving. Essentially, people have sins and the payment for those sins is Jesus coming.
Pretty short sighted if you ask me.
The context of verse 21 is the promise and directives to Joseph about what God is up to, and Joseph's responsibilities in that working. For some reason we miss the connection between this statement and the prophecy on which it is based. That prophecy, according to Matthew, does not use the name Jesus, but Immanuel. The implication is that Jesus will save his people from their sins by being God With Us. God has decided it was time to live with his people, and in so doing restore (save) them to full community with himself.
Being saved from sins has a number of connotations including the strictly juridical one. While it is true that we are "saved from our sins," the restoration of community with God is much farther reaching than that. In fact, the juridical view is essentially a consequence of having God elect to live with you.
There are others. Jesus tells us that his mission is to proclaim liberty to captives, sight for the blind, and setting at liberty the oppressed (Luke 4). We are saved from the sinful behavior of others; and we are set free from any perception that our flaws may make us something less than fully acceptable to God. To be saved from sins is to have a door open that others attempt to close because we are not "like them."
Jesus makes a few references to trees and fruit; that the fruit of a tree will reflect the quality of the tree. In making these sorts of comparisons, he draws our attention to the sorts of lives we live. For his hearers who considered themselves the people of God, these are challenges to live as though they actually understood their God. The attitude addressed here is the opposite of the Evangelical juridical one. Israel lived as though being the people of God was the entire point and that therefore they had it made. In the juridical sense, they were already selected and nothing could change that.
Jesus' comments about trees, fruit, wells, and water lead us to the conclusion that "saving us from our sins" includes as perhaps the most important point, saving us from a desire to live lives not in keeping with the God we claim to know.
"Saving us from our sins" then has a much more expansive import that includes God living with us, inviting us to live in community with him, a community characterized by mercy and grace among members, and members who want and who come to live lives of giving rather than getting. We are offered, literally, Heaven on Earth if we would just accept it.
The coming of God as human ushers in a worldview often missed by Evangelicals and other denominations. We often remember that we are "saved" by this baby, but we too often miss the invitation this child offers.