We read about the event you are referring to in Matthew 3:13-15, “Then cometh Jesus from Galilee to Jordan unto John, to be baptized of him. But John forbad him, saying, I have need to be baptized of thee, and comest thou to me? And Jesus answering said unto him, Suffer it to be so now: for thus it becometh us to fulfil all righteousness. Then he suffered him.” 

You are so right in stating that Jesus had no sin.  We read in God’s Word that the Lord Jesus:

• Knew no sin. (2 Corinthians 5:21).
• Did no sin. (1 Peter 2:22).
• There was no sin in Him. (1 John 3:5).

While the mode is probably the same, the purpose of John’s baptism is different from the purpose we observe today.  Today, believers on the Lord Jesus Christ are baptized to identify with the Lord in His death, burial, and resurrection.  We read in Romans 6:3-4, “Know ye not, that so many of us as were baptized into Jesus Christ were baptized into his death? Therefore we are buried with him by baptism into death: that like as Christ was raised up from the dead by the glory of the Father, even so we also should walk in newness of life.”  We read the words of John the Baptist in Matthew 3:11, “As for me, I baptize you with water because of [your] repentance [that is, because you are willing to change your inner self—your old way of thinking, regret your sin and live a changed life]…) (Amplified Bible).  Those who came to John for baptism were showing forth their repentance for the sins they had committed and acknowledged that they deserved death for their sins.

So why did this sinless one, the Lord Jesus, want to be baptized by John the Baptist?  Of course, when the Lord desired to be baptized of John, he refused, telling the Lord that he needed to be baptized of Him.  He understood the sinlessness of the Lord Jesus.  Yet, the Lord insisted that John baptize Him.  For what purpose?  I will quote from A.C. Gaebelein, “Then the King entered upon His public ministry. He too entered the waters of Jordan and was baptized; not that He needed it, but TO SIGNIFY HE CAME TO BE THE SUBSTITUTE OF SINNERS AND TO TAKE THE SINNERS’ PLACE IN DEATH. In the baptismal scene we have His whole blessed work foreshadowed. He is the holy One, who needed no baptism for He had no sin. His death is foreshadowed when He went into Jordan; His resurrection, when He came out of the water; His ascension, when heaven was opened unto Him. The gift of the Holy Spirit is seen next, and this is followed by the declaration of Sonship. Into this the believer enters. We are dead and risen with Him; Heaven too is opened unto us and we have received the Spirit of Sonship.”

Indeed, Jesus had no sin. But he was about to enter into His great work and public ministry. It was proper that He should be set apart by his forerunner, and show His connection with him, and give His approval to what John had done. Baptism was not a requirement under the Old Testament Law.  Rather, it was a procedure that the Lord had appointed John to do.  The Lord’s baptism acknowledged John’s work publicly as being directed by God.

We then read in verse 15, “Then he suffered Him.”  John understood and baptized the Lord Jesus.  It is then we hear the voice of the Lord from Heaven in verses 16-17, “And Jesus, when he was baptized, went up straightway out of the water: and, lo, the heavens were opened unto him, and he saw the Spirit of God descending like a dove, and lighting upon him: And lo a voice from heaven, saying, This is my beloved Son, in whom I am well pleased.”  (CC)  (545.2)