What does Acts 19:5 mean?

The twelve men in Ephesus are devout followers of the message of John the Baptist. They understand that they are sinners, that their sin is an insult to God, and that they need to follow God more closely. To that end, they received "the baptism of John," meaning, they publicly announced they that agreed with John's message and resolved to live according to his teaching (Acts 19:1–4).

Now, Paul has explained that the man whom John promised would baptize "with the Holy Spirit and fire" (Luke 3:16) is Jesus of Nazareth. John taught that people needed to follow God by obeying Him. Jesus taught that no one could earn God's forgiveness on their own. Jesus died as the perfect sacrifice, paying for the sin of the world. He rose from the dead in an affirmation that God accepted His sacrifice (1 Corinthians 15:14–19). Those who entrust Him for the forgiveness of their sins receive the Holy Spirit as a seal of their salvation (Ephesians 1:13–14; 4:30).

To be baptized in Jesus' "name" means to publicly submit to His message and authority. It doesn't mean the sounds that we use to identify Jesus are a magical spell. To be baptized in Jesus' name does not contradict Jesus' command to be baptized in the name of the Father, Son, and Holy Spirit (Matthew 28:19). The men had already been baptized by John; they know who God is. They need to add Jesus to their understanding and loyalties.

The sequence here seems a bit backward; first Paul baptizes them, and then they receive the Holy Spirit (Acts 19:6). This is the same sequence Philip saw in Samaria (Acts 8:12, 14–17). It should be noted that ancient writers, even those making orderly accounts, did not always focus on exact chronology. The greater focus in any story was ensuring all the major elements were recorded. Today, "credobaptism" is the standard. That is, people should be baptized after they have decided to follow Christ and can verbalize what they believe. We don't know the particulars, here. Paul didn't usually baptize his own converts (1 Corinthians 1:14–17). He may have assistants who baptize the men before he lays hands on them.

In the early part of the spread of the church, most of the new believers were either Jews or Gentiles who attended the Jewish synagogue and followed the Jewish God. When they came to an understanding and acceptance of Jesus' offer of salvation, they were baptized immediately (Acts 2:41; 8:12, 36–38; 9:18; 10:47–48; 16:15, 33; 19:5). Since these men had received the baptism of John, they know the basics. Today, people often become Christians and receive the Holy Spirit with less understanding of who Jesus is and who the Trinity is. Baptism typically follows a time of instruction. In that way, the new believers publicly align themselves as a Christian and the church publicly accepts them when they understand what being a Christian really means.