Text in red are my additions.
Luk 3:15 And as the people were of opinion, and all were thinking in their hearts of John, that perhaps he might be the Christ:
And the people were of opinion. (In the Greek πζοσδοκου̃ντες, suspecting, expecting, as Vatablus renders it—when the people were hoping, or were in suspense with hope, desire, and expectation), and all were thinking in their hearts of John, that perhaps he might be the Christ, or not—the Messiah promised to the fathers, and so eagerly expected by all the Jews at this particular time when the sceptre had passed from Judah, and Daniel’s seventy weeks, the sign of Christ’s coming, were fulfilled. As the people, then, were spreading this report about John, the chief men of the Jews at length sent messengers to him to ask him whether he were Christ (Joh_1:19). Such was the holiness of John. So S. Ambrose, Bede, and others explain.
Luk 3:16 John answered, saying unto all: I indeed baptize you with water: but there shall come one mightier than I, the latchet of whose shoes I am not worthy to loose. He shall baptize you with the Holy Ghost and with fire;
John answered, saying unto all: I indeed baptize you with water: but there shall come one mightier than I, namely, the Messiah.
The rest which Luke here adds has been explained on Matt 3:11.
Here is what Lapide wrote in Matt 3:11~I indeed baptize you, &c. These words must not be connected with what precedes, nor were they spoken immediately afterwards by John. But they were spoken as suitable to an occasion of which S. Luke gives an account and explanation (Luke3:15-16): “And as the people were of opinion, and all were thinking in their hearts of John, that perhaps he might be the Christ: John answered, saying unto all: I indeed baptize you with water: but there shall come one mightier than I, the latchet of whose shoes I am not worthy to loose. He shall baptize you with the Holy Ghost and with fire” From the sanctity of his life and the fervour of his preaching, and from his baptizing, the people suspected that John was the Messiah, or the Christ. For none of the other prophets, except John and Ezekiel, had made use of baptism. (See Ezek. 26, where he foretold that baptism would be a sign of Christ: “I will pour upon you clean water, and you shall be cleansed from all your filthiness.”) John therefore puts an end to this suspicion, and declares that he is not the Christ, but the forerunner and indicator of Christ, and that his baptism was a prelude to the baptism of Christ, and a preparation for it.
So he says, “I indeed baptize you in,” or “with water,” that is, with water only. This is a Hebraism, for the Hebrews denote the instrument by the preposition or letter ב, or in, which is understood in Latin. So the Hebrew said: במים, bammayim “in,” or “with water, unto repentance,” that I may stir you up to repentance, and that I may prepare you by corporeal ablutions for the washing of the soul to be received in the baptism of Christ. The baptism of John therefore was a profession of penance. Whence those who were about to be baptized by him confessed their sins, not that there was thereby a condonation of their faults; for this they were to wait for from Christ, by means of His baptism and true contrition .
Morally: Origen says, “Preachers are here warned not to allow themselves to be too much praised or honoured by the people, but to suppress these praises and honours, and refer them to Christ, lest by reason of their pride they be deprived of them by Christ.”
He will baptize you with the Holy Ghost and with fire. Both the person and the power of the coming one is greater than John and his activity. Holy Spirit (Ghost) and fire are sometimes taken as referring to Christian baptism, or to Pentecost, however, given the judgment motif, it is probably better to understand Holy Spirit here as meaning both Spirit and wind. Both the Greek and Hebrew words for spirit can also mean wind. Wind and fire are two OT images for Judgment: For wind see Psalm 1:4; Psalm 18:42; Hosea 13:15. For fire see Isaiah 26:11; Jeremiah 4:4.
Luk 3:21 Now it came to pass, when all the people were baptized, that Jesus also being baptized and praying, heaven was opened.
Now it came to pass. Indicates the importance of what is about to be related, concerning which see Acts 1:22, Acts 10:37-38. The close connection St Luke draws between our Lord and the people should not be lost sight of: when all the people were baptized…Jesus also being baptized. This connection recalls the angel’s words to the shepherds: Fear not; for, behold, I bring you good tidings of great joy that shall be to all the people (Lk 2:10). It prepares for the genealogy of our Lord which Luke traces back to Adam, the common ancestor of all people (Lk 3:23-38). It prepares for the Lucan presentation of Jesus as moving among the outcasts and sinners. Finally, it becomes an important theme in the apostolic preaching (Acts 13:24-25). In the more immediate context it helps highlight who Jesus is (verse 22), building upon the information given in the first two chapters of Luke. The one who closely associates with sinners, even to the extent of being baptized as they were, is the one singled out by God as My beloved Son. In thee I am well pleased. We begin to see what it means that Jesus was born to be God’s salvation, which thou hast prepared before the face of all peoples: a light to the revelation of the Gentiles and the glory of thy people Israel (Lk 2:30-32). the message will achieve a crescendo on Calvary when the thief hears these words: Today, thou shalt be with me in paradise (Luke 23:43). At the same time the distinction between Jesus and the people implies that a decision has to be made for or against him, as will become obvious as the Gospel proceeds, but it is a theme we have already come up against: Behold this child is set for the fall and for the resurrection of many in Israel and for a sign which shall be contradicted (Lk 2:34). Just as there were those who would not accept John’s baptism in an honest and forthright manner (Lk 3:7-9), so to some will not accept Jesus, his work and message (Lk 6:43-49, Lk 7:29-34, Lk 13:1-9; Acts 13:38-52, etc.).
Luk 3:22 And the Holy Ghost descended in a bodily shape, as a dove, upon him. And a voice came from heaven: Thou art my beloved Son. In thee I am well pleased.
The Holy Ghost descended in a bodily shape, as a dove. The reference to a bodily shape is probably meant to convey the reality of the event. It is interesting to note that the risen Jesus has to insist upon the reality of his risen body: See my hands and feet, that it is I myself. Handle, and see: for a spirit hath not flesh and bones, as you see me to have (Lk 24:39). The phrase as a dove should not be taken as a reference to the bodily shape the Holy Spirit took, i.e., that it descended in the form of a dove, rather, as a dove connects with the descent, i.e., the Holy Spirit descended as a dove descends. The bodily shape may or may not have been dove-like, at least this is the view of some scholars.
Thou art my beloved son. In thee I am well pleased. The words spoken are a conflation of Ps 2:7, a kingly enthronement psalm, and Isaiah 42:1, a suffering servant song. “The two text join sovereignty and service” (Fred B. Craddock, Interpretation Series: Luke, page 51).
The pronoun thou is emphatic, emphasizing Jesus alone. A contrast is being drawn between Him and all the people he will serve. The emphatic pronoun prepares for out Lord’s application of Isaiah 61:1-2 and Isa 58:6 to himself in Luke 4:18-19 (note the threefold use of “me”). The designation of Jesus as God’s beloved son builds upon the angel’s words in Lk 1:35~the Holy which shall be born of thee shall be called the Son of God. The designation of son likewise prepares for the genealogy which will identify Christ as son of Adam, son of God (Lk 3:38). Also, it prepares for the trial Jesus is about to undergo from Satan: If thou be the son of God, command these stone to become bread.