"'That is why the Jews seized me in the temple courts and tried to kill me.'" NIV translation
The Jewish leaders believed that they and only they were God's chosen people and so they did not see a place in the family of God for Gentiles. Paul explains that that was the reason that they were out to kill him.
"'But I have had God's help to this very day, and so I stand here and testify to small and great alike. I am saying nothing beyond what the prophets and Moses said would happen - that the Christ would suffer and, as the first to rise from the dead, would proclaim light to his own people and to the Gentiles.'" NIV translation
Paul explains that God has protected him against the schemes of those that were accusing him and we can take comfort in the fact that God will protect us too. He also reminds them that he is not saying anything different than what the Old Testament prophets had said.
There are many who do not understand that the Old Testament is all about the coming of Jesus Christ. We study the Old Testament because it helps us to understand more about Jesus Christ and the nature of God the Father.
"At this point Festus interrupted Paul's defense. 'You are out of your mind, Paul!' he shouted. 'Your great learning is driving you insane.'" NIV translation
At this point, it looks like things are going bad for Festus and he is possibly going to be blamed for this mess. Therefore, he jumps into the conversation and accuses Paul of being insane. That is a normal pattern in that, when people cannot disprove your words, they will resort to questioning your mental capacity. Jesus was accused of being out of His mind and now we see the same thing happening to Paul. If you are faithfully serving the Lord, you can expect the same thing to happen to you.
"'I am not insane, most excellent Festus,' Paul replied. 'What I am saying is true and reasonable. The king is familiar with these things, and I can speak freely to him. I am convinced that none of this has escaped his notice, because it was not done in a corner. King Agrippa, do you believe the prophets? I know you do.'" NIV translation
It is good to notice how Paul handled Festus in that he reacted in kindness but then turned right back to address the king. It is easy to get distracted from the work of the kingdom by the actions of others but we must be disciplined and focused on the things that matter.