Verse 5: To whom we gave place by subjection, no, not for an hour; that the truth of the gospel might continue with you. Paul's response was instant in that he says they did not yield to what these pretenders were saying not even for an instant. His point is that anything added to grace is no longer grace, i.e., Grace + Anything = Works. This is the other gospel that he was referring to in Galatians 1:8. This other gospel is what has destroyed the church in my opinion.
Verse 6: But of these who seemed to be somewhat, (whatsoever they were, it maketh no matter to me: God accepteth no man's person:) for they who seemed to be somewhat in conference added nothing to me: He is referring to the Counsel in Jerusalem and his meeting with the Twelve. His point is that they had nothing to add or offer in regard to the mystery that he shared with them. If anything, he shared things with them that they did not know, i.e., the mystery. He essentially said the same thing in 1:16.
Verse 7: But contrariwise, when they saw that the gospel of the uncircumcision was committed unto me, as the gospel of the circumcision was unto Peter; In the end, they all realized that what Paul had received was for the Gentiles and it was not the same message that Peter had for the Jews. Where I believe many go astray today is that they believe one of two things. First, the only thing that Paul shared with them was that the Gentiles were now included in God's plan. The problem with that view is that the Gentiles have always been in God's plan. It is just that he wanted to use the Jewish nation to do it and they refused.
Second, that Paul's revelation was simply that the Jews, now that they are Christians, should stop keeping the Law. Nope. Under the Gospel of the Kingdom, the Law was required. The bottom line was that Paul's message of grace was for the Gentile apart from the Law. Of course, after the rejection of the kingdom offer was complete, i.e., 70AD, it would be the only way of salvation to Jew and Gentile. Also, I do not believe that Paul fully grasped that at this time because progressive revelation was still happening for him.