Why did Paul want to prove to the Jerusalem Jews that Gentiles were free to eat anything?
Paul wanted to maintain the testimony of the Gospel of Grace by proving to the Jerusalem Jews that Gentiles are free to eat whatever they want and are no longer under the dietary laws. He opposed Peter's withdrawal from eating with Gentile converts, which compromised that freedom and the truth of the gospel.
Les explains that the issue arose because, before certain men came from James in Jerusalem, Peter ate with Paul and those Gentile believers regardless of what the meal was. But after Jewish people came up from Jerusalem, Peter became scared and withdrew, separating himself from the Gentiles because he feared the men from Jerusalem, not God. To maintain the testimony of Paul's Gospel of Grace, Paul needed to prove to these Jews from Jerusalem that believers are free to eat whatever they want, are no longer under the dietary laws, and have been set free from all that. Instead, Peter gave in and refused to go in and eat with Paul's Gentile converts. Les says Paul got real upset with Peter and called him two-faced, since Peter had no problem eating with the Gentiles when no one from Jerusalem was there but changed as soon as they came.