"After these things, during the reign of Artaxerxes king of Persia, Ezra son of Seraiah, the son of Azariah, the son of Hilkiah, the son of Shallum, the son of Zadok, the son of Ahitub, the son of Amariah, the son of Azariah, the son of Meraioth, the son of Zerahiah, the son of Uzzi, the son of Bukki, the son of Abishua, the son of Phineas, the son of Eleazar, the son of Aaron the chief priest - this Ezra came up from Babylon. He was a teacher well versed in the Law of Moses, which the Lord, the God of Israel, had given. The king had granted him everything he asked, for the hand of the Lord his God was on him." NIV translation
The phrase "After these things" speaks of the rebuilding of the temple that we saw in chapter 6. We are introduced to Ezra and his heritage as a priest as his family tree is traced back to Aaron who (along with Moses) had led God's people out of captivity in Egypt. Before the temple was rebuilt, there would not have been a place for the priest to perform the sacrifices and so he stayed behind in Babylon teaching the people the Law of Moses. Finally, we are told that the favor of God was upon him and this caused the king to give him all that was needed for his work.