"When Silas and Timothy came from Macedonia, Paul devoted himself exclusively to preaching, and testifying to the Jews that Jesus was the Christ. But when the Jews opposed Paul and became abusive, he shook out his clothes in protest and said to them, 'Your blood be on your own heads! I am clear of my responsibility. From now on I will go to the Gentiles.'" NIV translation
When the Jews would not listen and became abusive, Paul moved on to share the good news with the Gentiles. You may say that Jesus would never turn His back on anyone but Paul's actions are in line with what Jesus said when He sent the disciples out into the world (Matthew 10 and Luke 10). This can serve to remind us that we are not called to save others as only Jesus can do that and God must call them but we are to boldly declare the Gospel just as Jesus and Paul did. When people reject the message and become abusive, we too are released from responsibility.
So, Paul left the synagogue and went next door to the house of a believer. Paul did not leave the synagogue in defeat as we see that even the synagogue ruler and his entire household believed.
The local synagogue of this day was overseen by elders and they often appointed a ruler.
This ruler was a layman (not a priest) and he cared for the building and its contents as well as selecting the people who would take part in the service on the Sabbath. So, when Paul left the synagogue, we see that the ruler went along and they continued to meet at this house for worship and learning the Word of God. This would be what we consider today a "house church" and in many areas of the world it is the only way that believers get together for worship.
"One night the Lord spoke to Paul in a vision: 'Do not be afraid; keep on speaking, do not be silent. For I am with you, and no one is going to attack and harm you, because I have many people in this city.' So Paul stayed for a year and a half, teaching them the word of God." NIV translation
Paul must have been wavering in his mind about sharing the gospel here because the Lord spoke to him to reassure him. This can remind us that we are all human and, like Paul, sometimes we may question whether we want to do as the Lord has shown us or give up. We can take comfort in the fact that even Paul went through this but the Lord comforted him and will do so for us as well. Paul was also reassured in that the Lord told him that he had "many people in this city". We can also take comfort in the fact that God is in control and He knows his people. Even though Paul did not know how many people from that city God was calling to himself, God knew from the beginning of time.