In this passage, Jesus was visiting Nazareth, the place where he had grown up. Everyone knew who he was and who his family was, and so they were amazed at how well he spoke. However, it seems they also expected him to do some powerful miracles to show that he really was important, and he was not willing to do this. In another visit to Nazareth it says:
And they took offense at him. But Jesus said to them, “A prophet is not without honor except in his hometown and in his own household.” And he did not do many mighty works there, because of their unbelief. (Matthew 13:57-58)
The people thought that because their nation, Israel, was the chosen nation of God, that Israelites should be treated specially by God’s prophets. In the same way, they thought Nazareth should be treated specially because they knew more about Jesus than the other towns he had done miracles in. In Luke 4, Jesus responds to this expectation by giving examples of prophets who did not do mighty works in their own town or even for the nation of Israel. Elijah went to a widow in Zarephath rather than one in Israel, while Elisha healed the foreign general Naaman rather than healing Israelite lepers. Jesus was giving examples where the “inferior” Gentiles had been shown miracles that the chosen people of Israel had not been shown.
A very similar example is shown in Acts 22:21-22, where the Jews were quite willing to listen to Paul until he said that he had been sent to the Gentiles, and then they shouted that he didn’t deserve to live. Going to the Gentiles seemed to challenge everything that made the nation of Israel important and special, so their pride in their national heritage led them into a murderous rage.