Why did Peter finally agree with Paul at the Jerusalem council?
Peter finally agreed because the Holy Spirit awakened him to recall that God had earlier sent him to the house of Cornelius and showed him He could save a Gentile. That prior revelation convinced him Gentiles are saved by grace, so he concluded Paul was right and the council gave in.
Les explains that Paul went to Jerusalem and withstood them, insisting his pagan converts recently come out of idolatry did not have to keep the Judaic Law or practice circumcision because they were now saved by grace through faith plus nothing. The great conference went on a good part of the day until finally Peter, by the work of the Holy Spirit, woke up and remembered that twelve years before God had sent him to the house of Cornelius and shown him He could save a Gentile. Peter then said he guessed Paul was right, and they gave in, as recorded in Galatians chapter 2. Les also notes that James, Peter (Cephas), and John—the inner core of the twelve—perceived the grace given to Paul and gave him and Barnabas the right hands of fellowship, agreeing he should go to the heathen.