Why were the Jewish believers in Jerusalem upset with Paul and Barnabas?

The Jewish believers in Jerusalem were upset because Paul and Barnabas were bringing Gentiles into the church at Antioch and offering them God's salvation. This inclusion of Gentiles was anathema to the Jews of that time, causing consternation, so they sent men to investigate and summoned Paul and Barnabas to Jerusalem.

Les explains that the Jewish believers of Jerusalem had gotten all shook up over the fact that Paul and Barnabas were bringing Gentiles into the church up at Antioch. Consequently, they sent men from Jerusalem to check it out, and indeed it was true. It caused so much consternation that they requested Paul and Barnabas come down to Jerusalem, which Les calls calling them on the carpet, because they were bringing Gentiles in and offering them God's salvation. Les notes that this was just anathema to the Jew of that time.

Acts 15:1

And certain men which came down from Judaea taught the brethren, and said, Except ye be circumcised after the manner of Moses, ye cannot be saved.

What were the Judaizers teaching the Gentiles in Antioch that Paul had to oppose?

They taught Gentile believers at Antioch that unless they were circumcised according to the manner of Moses they could not be saved, and that it was necessary to circumcise Gentile converts and command them to keep the law of Moses.