"Paul answered, 'I am now standing before Caesar's court, where I ought to be tried. I have not done any wrong to the Jews, as you yourself know very well.'" NIV translation
Paul knew that God had told him he would be going to Rome and he also realized that Festus was just another politician. He expected Festus to do what was best for Festus and not always what was in the interest of justice. Therefore, he refuses to go to Jerusalem for trial by reminding them that it was not necessary.
"If, however, I am guilty of doing anything deserving death, I do not refuse to die. But if the charges brought against me by these Jews are not true, no one has the right to hand me over to them. I appeal to Caesar!" NIV translation
Some have used this verse to describe Paul as being afraid but we can clearly see that fear was not his motivation. He was not afraid to die as he knew that he would then be with Jesus forever. We must remember that Paul had known for quite some time that he was going to have the opportunity to tell rulers about Jesus Christ.
He knew that God wanted him to go to Rome and he was willing to obey. As a Roman citizen, he had the right to appeal to Caesar and in so doing he was in the center of God's will for his life.
"After Festus had conferred with his council, he declared: 'You have appealed to Caesar. To Caesar you will go!'" NIV translation
After consideration, Festus realized that he could get rid of the problem of Paul and not have any blame with the Jews. Therefore, he agreed to send Paul to Rome for trial.
"A few days later King Agrippa and Bernice arrived at Caesarea to pay their respects to Festus. Since they were spending many days there, Festus discussed Paul's case with the king. He said, 'There is a man here whom Felix left as a prisoner. When I went to Jerusalem, the chief priests and elders of the Jews brought charges against him and asked that he be condemned.'" NIV translation
In this passage, we see that Festus was just another politician. His boss (King Agrippa) showed up and he explained that it was Felix's fault for the situation with Paul. It did not matter how Paul had been delivered to him for justice in that he could have still done the right thing. Instead, he shifts the blame to the governor before him. He also blames the Jews because they had brought the charges even though he could have simply dismissed them. As Christians, this is the opposite of what we are told to do in preparation for the attacks of the enemy (see Ephesians 6).