The Baptism of Jesus introduced him into his ministry. Jesus approached John in humility. He had no need for baptism, as he was sinless. John’s initial reaction was out of humility, but the gentle insistence of Jesus allowed him to understand that his baptism of Jesus was an act of service to God, because Jesus requested it.
The trinity is present at the Baptism: Jesus the son is baptized, God the father announced Jesus is His son, and the Holy Spirit descended on Jesus. Believers are baptized, and adopted as children by God, and receive the Holy Spirit.
What does it mean to “fulfill all righteousness”? Jesus fulfilled all righteousness by being obedient to God’s plan, which is always righteous. He also set an example for us to follow, that we humble ourselves and submit to God. By accepting baptism, Jesus acknowledged the sins, our sins, for which he would atone. He was not in need of baptism, nor was he deserving of death, but he subjected himself to both, for our sake.