The issues over which Paul expresses anger to the Gentile Galatians are predominantly two-fold: (1) the authority of the Spirit versus the authority of the Law and (2) the place of the Gentile Christian in the Jewish Christian community.  Jesus Christ first introduced the concept of an in-filling Spirit (synonymous with the spirit of Christ) whose Presence fulfilled the requirements of the Jewish Law. As a new spiritual and theological concept, there was a great deal of confusion and controversy surrounding precisely what the impact of the Spirit was on various aspects of conversion. In this instance with the Galatians, Paul's outrage was that the Galatians were falling under an influence that gave the authority of the Law precedence over the authority of the Spirit, while Paul taught the opposite.
 Jews who began to follow the way of Jesus Christ had been fulfilling the requirements of the Jewish Law throughout their lifetimes. For these, it was inconceivable that Gentiles could convert to the way of Christ without first converting to the Laws of Judaism as had always been the way with Gentiles who wanted to become Jews. A group of proponents for such a Law related conversion were advocating adherence to the Law among Gentiles so they could enter into Judaism according to the traditional way. Paul found this to go contrary to the new Law of the Spirit and, since the debate he had had with John and the other apostles had gone in his favor, Paul was adamant in pressing his point that the Law was immaterial to the conversion of Gentiles to the way of Christ.
Keep in mind, Paul's ministry quickly expanded to bringing the teachings of Jesus to non-Jewish areas. This was highly controversial, and there was great discussion and division regarding what was required to be considered a member of the new churches that Paul and the other apostles were founding. There were many believers who strongly felt that Jesus had been raised as a Jew, living under and intimately familiar with all the requirements and teachings of the Torah. These individuals could not conceive that God could love and accept as true believers those who did not conform to all the ancient laws. Paul struggled, in Galatia and elsewhere, to oppose this view; to clearly state that God's love and forgiveness was based on a person's true faith in Jesus Christ, not on that person's conformance to Jewish laws that were now superseded.
One of the central issues that the early church struggled with, and not just with Galatia, was how to interpret the Jewish law in the light of Christ and his teaching and death. Many argued that strict adherence to the Jewish law, including such activities as circumcision, was necessary. This is why Paul is quite harsh in his condemnation of such people. In Chapter Three he refutes such arguments, saying that because of Christ we have no need to follow the law. In particular, you may wish to look at verses 23-25, which states his argument about the purpose of the law and why it no longer needs to be followed:
Before this faith came, we were held prisoners by the law, locked up until faith should be revealed. So the law was put in charge to lead us to Christ that we might be justified by faith. Now that faith has come, we are no longer under the supervision of the law.
Thus it is that now faith has arrived through the person of Jesus Christ, adherence to the law is no longer necessary.
Paul's anger with the Galatians in the New Testament, Galatians, Chapter Three, is based upon their conversion from Judaism to following the teachings of Christ, and then trying still to follow the Jewish laws.
Paul points out to the people of Galatia that when they chose to follow Christ and his teachings, they put the old laws behind them—the "means of the flesh." He reminds them that the old laws never fed their spiritual life—that was not until they accepted the Holy Spirit. He asks them that after all they have suffered in the name of Christ, are they are willing to give that up by returning to the old ways and following the Jewish laws again?
Paul also reminds them that God gave them the power of the Holy Spirit, and has worked miracles in their midst because they have attempted to obey God's laws—not the old laws. And that the people must continue to trust God. (Galatians 3:1-5)
Paul reprimands the Galatians in Chapter 3 because of the influence of the so called "Judaizers," who insisted that Gentile Christians must follow the Jewish law. Because the early Christians were all Jewish and had been circumcised and compelled to follow the law, they believed that the Galatians should do the same. Paul's argument is that one is freed from the law when one becomes a Christian. The following quote from the third chapter illustrates the point:
Did you receive the Spirit by the works of the law, or by believing what you heard? Are you so foolish? After beginning by means of the Spirit, are you now trying to finish by means of the flesh? Have you experienced so much in vain—if it really was in vain? So again I ask, does God give you his Spirit and work miracles among you by the works of the law, or by your believing what you heard?
In other words, they have heard the word of Christ preached to them; but some still stubbornly held to the law; including the law of circumcision and the dietary laws. Since Christ had redeemed Christians by his death on the cross, the purpose of the law was fulfilled, and there was no longer any need to be bound by it. To insist on strict observance of the law was to Paul, folly. For that reason, he speaks rather harshly to the members of the Galatian Church.
One of the major issues that Paul was addressing in many of his epistles is the question of how Jewish the Christian faith was going to be. The Christian faith had, after all, come out of Judaism and there were questions going on as to how much the Christians should follow Jewish law. In Galatians 3, Paul is scolding the Galatians for believing too strongly in the law. Paul does not want the people to believe that they are supposed to put their faith in the law. Instead, he wants them to put their faith in the spirit. He tells them that they cannot receive the spirit of God through the law.
So, the main point of this chapter is that the Christians should not put their faith in following the law. He is mad at them because they are doing just that; they are caring too much about following the law.