Acts 15:31-35
Acts 15:31 (KJB)
Which when they had read, they rejoiced for the consolation.
Consolation - Encouragement or comfort
It is not stated who read the letter but the fact was they received the answer and when it was read, they had rejoiced because of the encouragement they received from it. Now they know that salvation is by grace alone without any additions to of the law as a requirement for salvation. Now that they had the answers they were looking for, they would now be able to reject the Judaizers and show them and the others that their teaching is false by laying an extra burden on the believers, a burden which God never gave.
Acts 15:32 (KJB)
And Judas and Silas, being prophets also themselves, exhorted the brethren with many words, and confirmed them.
Not only did the church have the written letter but Judas and Silas were prophets and began to confirm the brethren orally. This was still the time when the Bible was not yet completed so they were able to receive exhortation directly from the Holy Spirit. As they received instruction from Him, they relayed it unto the church. Since there was probably an air of confusion in the church because of the Judaizers bringing in the false teaching, Judas and Silas had probably taught them and helped bring back a semblance of order to their beliefs by removing the concept of keeping the law of Moses for salvation. This was Godís method over hundreds of years as He would raise up a prophet and send him to the nation of Israel when they were being disobedient. Here, though, it was a different story, the church was young and needed to know about this subject, so they were given a letter and two men who prophesied for them the truth of the Gospel. By these two methods, the church was once again able to get back to the ministry of the word. Unfortunately, these incursion of false teachings can stultify a church and bring its ministry to a halt until these issues are resolved.
Acts 15:33 (KJB)
And after they had tarried there a space, they were let go in peace from the brethren unto the apostles.
Space - Time or respite
The group from Jerusalem had stayed there for a while, no doubt to rest and to help teach because it sometimes takes time and effort to unseat false teachings when they have become not only entrenched but only introduced. After sufficient time had passed and the threat of these false teachings were passed, the brethren were able to leave and head back to the Jerusalem church and not just to the Apostles.
Acts 15:34 (KJB)
Notwithstanding it pleased Silas to abide there still.
Judas had gone back to Jerusalem but for some reason Silas chose to remain at Antioch. No doubt he wanted to stay to make sure if the Judaizers return to Antioch at some future time, he will rebuff them and their false teachings so the church would have a fighting chance. He also stayed to help with the teaching and discipline.
Acts 15:35 (KJB)
Paul also and Barnabas continued in Antioch, teaching and preaching the word of the Lord, with many others also.
Since the church in Antioch was growing, they needed all the teachers they could find and that is why Paul and Barnabas had stayed behind to continue teaching in the church and also to help train other believers for teaching and leadership roles. In the church, Christ had given pastors, evangelists, and prophets to bring about maturity among the believers. It was for them to train the believers so they would go out from the church and evangelize and disciple those who became saved. It is not stated how long Paul and Barnabas stayed there but probably the next verse will give us the reason why Paul wanted to stay in Antioch, beside teaching and training the believers.