Matthew's account of Jesus' baptism is a rewriting of traditional materials so as to explain why Jesus "the One Mightier" had to be baptized by John. Below is a table showing the accounts of Matthew, Mark and Luke in the NAB translation (click on the image for a larger, printable view)
Mark's account of Jesus' baptism is straightforward and simple. Jesus from Nazareth goes to the Jordan to have himself baptized by John. There he receives a three-fold revelation: the heavens being torn apart, the Spirit descending and a voice that said "You are My Son, the Beloved; in you I am well pleased."
Luke builds on this Marcan material emphasizing two things: (a) the revelation occurs in a moment of intense attention to God (while he was praying) and (b) with the genealogy that immediately follows (Luke 3:23-38), he shows how the baptism is an act of solidarity with all men. The opening of the heavens is attenuated in Luke since in his account of the birth of Jesus, the angels already proclaimed Jesus as heaven-sent. Mark, on the other hand, writes of the tearing apart of the heavens which will correspond with the tearing of temple veils later on at Jesus' death.
Matthew too builds on the account of Mark by inserting a dialogue between the Baptist and Jesus and by showing that the voice from heaven makes a public presentation of Jesus as God's Son, the Beloved in whom He is well pleased.
When Matthew presented the ministry of John, he also gave a summary of the Baptist's message: "Repent for the kingdom of heaven is at hand (3:2)". He also made it clear that his baptism is a "baptism with water for repentance". For members of his community who already know Jesus as one who will save his people from their sins (1:21), the baptism of repentance that Jesus undergoes can be perplexing. It would imply that Jesus was a sinner. To explain why Jesus had to be baptize, Matthew inserts vv. 14-15 which is a dialogue between John and Jesus.
14. John would have prevented him, saying,
"I need to be baptized by you, and do you come to me?" 15. But Jesus answered him,
"Let it be so now;
for it is proper for us in this way to fulfill all righteousness."
Then he consented. (NRSV)
The dialogue is framed by the verbs for "preventing" (13) and "allowing" (15). Jesus had presented himself to John for baptism but John would have none of it. He only allows Jesus to descend into the waters after being convinced that it was the right thing to do. A similar case is found in the account of the annunciation to Joseph who would have divorced Mary if an angel had not explained to him the maiden's situation.
The Baptist's objection gives voice to what the readers of the gospel already know fully well about Jesus. Matthew's insertion of the birth narrative indicates that his readers have already become familiar with an elevated Christology that puts Jesus in the realm of God where there is no sin. Jesus' reply explains it all: "Let it be so now for it is proper for us in this way to fulfill all righteousness."
"Righteousness" is a key word in Matthew's gospel (3:15;5:6.10.20;6:1.33;21:32). Here it means "what is right in the eyes of God". Jesus' baptism is part of a design which we can only guess at from the perspective of the whole gospel. In Mark, Jesus' baptism is an act of solidarity with the whole of Israel, the first part of a process where he undoes Israel's sin of disobedience. In Matthew, this Marcan insight is deepened in the account of the temptation in the desert.
Matthew's narrative takes on a particular hue after Jesus' baptism. Here the revelation about Jesus takes on a public character. Matthew moves 
, making its object the Spirit descending like a dove; he frames the vision and the voice with "behold" (16.17) underlining the revelatory aspect of the scene; he adds the verb 
to describe the "voice". While Luke and Mark keep the revelation private, Matthew turns the scene into a formal presentation of Jesus. Matthew had already shown Mary presenting her son to representatives of the Gentiles. Now the presentation is made to whoever reads or hears the gospel by a heavenly voice. Previously, the verb 
was used for prophetic voices (see its use in the birth narratives). Now it is used for God's voice.
This is My Son, the Beloved
in whom I am well pleased.
As in Mark, this message will be repeated in the Transfiguration (Mt. 17:5). It is a combination of Messianic titles based on Psalm 2:7 and Isa. 42:1. In the light of the birth narratives the title "My Son" acquires a new nuance. It no longer just refers to a Davidic-Messianic lineage but to a heavenly one. The line from Isa. 42:1 -- "...in whom my soul is pleased..." -- identifies Jesus as the Suffering Servant upon whom God has put His Spirit, and whom He sets up as a "covenant to the people, a light to the nations" (Isa. 42:6).
An element in this public revelation is the Spirit of God ("Spirit" in Mark, "Holy Spirit" in Luke) which Matthew describes using two verbs: one for "descending" and another for "coming". The same Spirit which made Jesus' conception possible (1:20) now alights on him (3:16, NRSV) in a graphical presentation of Isa. 42:1. It is by this Spirit that Jesus will be led to the desert to face the Tempter as "Son of God".
