Luke 3:15-17 has been covered in the notes for Advent III Gospel Series C. Please refer to those notes. The main thrust for this day is the baptism of Jesus.
Look at parallels in Matthew 3:13-17; Mark 1:9-11, and John 1:32-34. John does not record the actual baptism. But all four evangelists record the descent of the Spirit. According to John the descent of the Spirit identified Jesus for the Baptist. Mark and Luke agree as to the words of the Father from heaven. (Note that the variant reading in Luke 3:22, taken from Psalm 2:7, is rejected by the translations and the better commentaries.) Matthew has the same words, except that they are put in the third, not the second, person. From this incident the Baptist testified: "This is the Son of God." John 1:34. Jesus' words in Matthew 3:15: "Thus it becometh us to fulfill all righteousness" are exceedingly important for our understanding of the baptism of Jesus.
Verses 21 and 22 comprise one sentence. It has a three-fold subject, and three aorist infinitives. "Jesus," "Holy Spirit" and "A voice from heaven (Father)"
This records, not a vision, but an actual sight. John 1:32 brings that out clearly. Matthew remarks that Jesus Himself saw the Spirit descend as a dove. "Like a dove" shows in what actually form the Spirit descended. All four Gospels make this observation. It is unmistakably clear. The words "You are My Son, etc.," of course, are the content of the voice from heaven.
Not only did the Holy Spirit come down on Jesus but it remained on Him. We are speaking of the incarnate Christ, the Messiah, now equipped to do His work as Messiah. This, of course, does not mean that He did not have the Holy Spirit before. On the thought compare Isaiah 61:1ff and Acts 10:38.
On two other occasions, in Luke 9:35 and John 12:28, the Father spoke from heaven with reference to Jesus.
"In Thee I am well pleased." "On Thee my favor rests." "I am delighted with You." The Father is not speaking of Jesus' divine nature. He is speaking of the incarnate Christ, the God-man.
With these words we see the relationship between the Father and the incarnate Christ. He fully approves of Jesus' person. The Father fully approves of Jesus' work for mankind. These notes do not go along with those commentators who see the idea of election or choice in this word.
The incarnate Son of God is beloved of the Father and the Father is well-pleased with Him. Everything that Jesus was and did is for me. Therefore, I am beloved of the Father and He is well-pleased with me.
Though Jesus was the sinless Son of God, when He was baptized He took the sins of the world upon Himself and in that sense had to be baptized. Look at Matthew 3:15 and John 1:29.
Here we have a clear teaching concerning the Trinity. The Trinity is always pictured to us as the saving God. All three persons are gracious and merciful to us in Christ Jesus. The Holy Spirit came in the form of a dove which indicates His graciousness and peace. By the way, the Spirit appeared in physical form both here and on Pentecost to assure us of His presence.
Luther: Heaven opens itself, which hitherto was closed, and becomes now at Christ's baptism a door and window, so that one can see into it; and henceforth there is no difference any more between God and us; for God the Father himself is present and says, This is my beloved Son.
Geldenjuys: Quotes Isaiah 64:1 and says, O that thou wouldst rend the heavens, that thou wouldst come down! And then he comments, In Christ Jesus heaven has been opened to us, the way has been paved for us to go to the Father's eternal home as His beloved children, saved by our Redeemer.
Arndt: Luke records a circumstance omitted by Matthew and Mark, that Jesus prayed after the baptism had been performed. Jesus stands before us as the obedient, humble Servant of the Lord, submitting entirely to the Father's guidance.
That is a good observation. Reformed commentators say that Jesus prayed as a model for us. No. Arndt is surely correct.
Arndt: It must be remembered that we are dealing with a story of the God-man, whose very person is more mysterious to us than any of the miraculous events that are reported from His life.