What does Acts 18:6 mean?

Whenever Paul enters a new town to share Jesus' offer of forgiveness, he first goes to Jews and Gentile God-fearers. If there is a synagogue, he goes there and shows how Jesus of Nazareth fulfills the prophecies about the Messiah in the Hebrew Scriptures: our Old Testament. Invariably, some Jews and a great number of Gentiles—in and out of the synagogue—believe. But Judaism is a community religion, and if the whole synagogue leadership doesn't choose to follow Jesus, Paul must leave.

This time, in Corinth, the synagogue ruler believes (Acts 18:8), but not enough of the rest of the leadership. Paul responds with a cultural act of dismissal and a reference to a prophet.

After the Jews returned from exile in Babylon, Nehemiah traveled to Jerusalem to encourage them to rebuild the wall around the city. While there, he discovered that the rich were abusing their power and using such unfair business practices that the poor had to sell their children into slavery. Nehemiah confronted the rich, and they agreed to stop charging interest on loans and to return the land and homes they'd taken (Nehemiah 5:1–12). Nehemiah writes, "I also shook out the fold of my garment and said, 'So may God shake out every man from his house and from his labor who does not keep this promise. So may he be shaken out and emptied.' And all the assembly said 'Amen' and praised the LORD. And the people did as they had promised" (Nehemiah 5:13). This parallels Jesus' command to the disciples that if a town refused to believe His message, they should "shake off the dust that is on your feet as a testimony against them" (Mark 6:11) as if even the dust they walked on was unworthy of Jesus' offer of salvation.

The line that their blood would be on their own heads has to do with who is responsible for whatever curse or hardship would fall upon them. Ezekiel the prophet served God in Babylon. God gave him very difficult messages to tell the exiles there, including that they would not return to Jerusalem anytime soon and that the temple would be destroyed. In Ezekiel 33:1–9, God explains that He chooses prophets to be watchmen for the people to listen to. If the people refuse to listen to the prophets, they will be responsible when God's judgment brings their death. If the prophet refuses to share the message God has given him, the prophet is responsible.

In the case of the Corinthian synagogue, Paul has shared the message Jesus commissioned him to; he fulfilled his responsibility. He is absolved of any liability for those who refuse to believe him.