What does Acts 19:3 mean?

Paul has just arrived in Ephesus and has met twelve men who know a lot about John the Baptist but not much about Jesus. John's ministry was largely around the Jordan River east of Jerusalem. He had been selected since inception to herald the coming of the Messiah (Luke 1:16–17). His personal message was one of repentance: that people should turn away from their rebellion against God and intentionally choose to obey Him, instead. To show their acceptance and devotion to John's message, people were baptized into "John's baptism."

We don't know where or by whom these men were baptized. Jewish men, no matter where they lived, were expected to go to Jerusalem to participate in the three main feasts each year; Jews who lived outside of Judea went far less often, but still tried to make some. By this time, John has been gone for decades. The men may have learned about John from Apollos, a Jewish man from Alexandria who devoutly followed John's message and preached it in Ephesus. The first time Paul came by Ephesus, he left behind Priscilla and Aquila who met Apollos and explained to him that John had been talking about Jesus of Nazareth. Apollos quickly realized he needed to follow Jesus, not John, and took that message to Corinth (Acts 18:24–28).

To be baptized into a "name" means to publicly affirm you follow the teachings of that person. "Name" doesn't just mean a series of sounds that identifies an individual. It includes the person's character, reputation, beliefs, and authority. The men had received John's baptism, meaning they committed to following his teaching. Now, they need to be baptized in Jesus' name to show they trust Jesus' message: that He is the Messiah who died and rose again for the forgiveness of their sins.