First, a bit of context. Isaiah 40-55 was written during the Exile in Babylon, when the nation of Israel was in captivity. It was a period of profound re-imagination of God and of Israel's character and purpose. There are four purple passages in this portion of the book of Isaiah (often called Second Isaiah or Deutero-Isaiah):
- Isaiah 42.1-7
- Isaiah 49.1-6
- Isaiah 50.4-9
- Isaiah 52.13-53
They build up a picture of the servant, who suffers, is a light to the Gentiles, establishes justice, is a king, sustains the weary, trusts God, etc.
Precisely who this refers to is unclear. Does it refer to a specific king? (Some have suggested that it might refer to Zedekiah, or rather implausibly Uzziah.) Does it refer to the whole nation of Israel? Does it refer to a Messiah? Does it refer to a particular future king? Could it refer to King Cyrus of Persia, elsewhere in Deutero-Isaiah known as God's "anointed"?
It is impossible to know for certain, and in fact it isn't terribly important to know exactly what the author(s) of the text intended. The more important thing is that the prophecy of Deutero-Isaiah has been understood throughout the Christian tradition as prophetically pointing to Jesus.
For instance, the Catechism of the Catholic Church quotes the precise verse you mention in saying:
In Jesus, the Law no longer appears engraved on tables of stone but "upon the heart" of the Servant who becomes "a covenant to the people", because he will "faithfully bring forth justice".(CCC, §580)
The other servant songs are also cited:
The Messiah's characteristics are revealed above all in the "Servant songs." These songs proclaim the meaning of Jesus' Passion and show how he will pour out the Holy Spirit to give life to the many: not as an outsider, but by embracing our "form as slave." Taking our death upon himself, he can communicate to us his own Spirit of life.(CCC, §713)
As to why it is the OT reading for today, the feast of the Baptism of the Lord... Well, you are quite right to pick up on the allusion to the verse in the Gospel passage. The words from Heaven are a combination of Psalm 2.7 and Isaiah 42.1 (especially clear when looking at the Septuagint version of these passages). It has traditionally been understood as demonstrating Jesus' identity as both King and Servant.