"What shall we do? They will certainly hear that you have come, so do what we tell you. There are four men with us who have made a vow. Take these men, join in their purification rites and pay their expenses, so that they can have their heads shaved. Then everybody will know there is no truth in these reports about you, but that you yourself are living in obedience to the law." NIV translation
So, Paul was accused of telling the Jews that they did not have to keep the law to be saved and this made the other Jews angry. The solution from the leaders in Jerusalem was for Paul to demonstrate obedience to the law. Paul was ok with obeying the law; he just knew that obedience to the law did not bring salvation because salvation was only possible through Jesus Christ.
"'As for the Gentile believers, we have written to them our decision that they should abstain from food sacrificed to idols, from blood, from the meat of strangled animals and from sexual immorality.'" NIV translation
The elders are saying that they have handled this question by sending these instructions to the Gentile believers.
By putting them under part of the law it satisfied the Jews. This decision was made at a meeting in Jerusalem described in chapter 15.
"The next day Paul took the men and purified himself along with them. Then he went to the temple to give notice of the date when the days of purification would end and the offering would be made for each of them." NIV translation
Paul agreed to take part in this ritual purification because he knew that it was not a matter of salvation. He knew that he did not need to do it to be saved but was willing to do so to reach others. This ritual is a picture of our repentance as the people had to be humble enough to let people know they had sinned (were unclean) by announcing it at the temple and that they wished to be cleansed (freed from the sin).
"When the seven days were nearly over, some Jews from the province of Asia saw Paul at the temple. They stirred up the whole crowd and seized him, shouting, 'Men of Israel, help us! This is the man who teaches all men everywhere against our people and our law and this place.'" NIV translation
So, even as Paul was obeying the law, he was seized and accused of leading God's people astray (being a false prophet). In fact, Paul was not doing what they accused him of; he was simply telling everyone that Jesus had died for their sins and been raised again. The charge of being a false prophet has been used throughout history by religious people who do not agree with someone.