Ezra does not name its writer
Unlike some Bible books, Ezra has no opening line that identifies its author. It begins with Cyrus, king of Persia, permitting Jewish exiles to return to Jerusalem and rebuild the temple, fulfilling what God had spoken through Jeremiah (Ezra 1:1-4).
So the clearest answer is modest: Scripture does not explicitly say, “Ezra wrote this book.” The title connects the book with Ezra, but a title is not quite the same thing as a signed author statement.
Now in the first year of Cyrus king of Persia, that the word of the Lord by the mouth of Jeremiah might be fulfilled, the Lord stirred up the spirit of Cyrus king of Persia, that he made a proclamation throughout all his kingdom, and put it also in writing, saying,
Why Ezra is the traditional author
Ezra was well placed to preserve this history. He was a priest in Aaron's line and a scribe, meaning a trained teacher and copyist of the Law of Moses (Ezra 7:1-6). King Artaxerxes sent him to Jerusalem with authority to teach God's law among the returned people (Ezra 7:11-14).
The strongest clue comes later in chapter 7. The account shifts from speaking about Ezra to recording his own response of praise to God (Ezra 7:27-28). In the following chapter, the first-person account continues as he describes gathering travellers and seeking God for a safe journey (Ezra 8:21-23). That makes good sense if Ezra wrote these portions, or if the book preserves memoirs he wrote.
Blessed be the Lord God of our fathers, which hath put such a thing as this in the king's heart, to beautify the house of the Lord which is in Jerusalem: And hath extended mercy unto me before the king, and his counsellors, and before all the king's mighty princes. And I was strengthened as the hand of the Lord my God was upon me, and I gathered together out of Israel chief men to go up with me.
The writer used several records
Ezra contains more than one kind of material. There are lists of returning families, correspondence between officials, royal commands, and reports about rebuilding the temple. At one point, Persian officials search the archives and find Cyrus's earlier decree concerning the temple (Ezra 6:1-5).
That does not weaken Ezra's connection with the book. Ancient historians often gathered trustworthy documents alongside personal memories and public events. Ezra may have assembled these sources himself, or an unnamed editor may have arranged some material around Ezra's records. The Bible simply does not settle every question about the final process.
Read Ezra for its larger story
The book's central message is clearer than its authorship details. God brought his people back from exile, enabled the temple to be rebuilt, and called the community to renewed faithfulness (Ezra 6:14-15). Ezra's own prayer shows a leader grieved by sin yet relying on God's mercy (Ezra 9:5-8).
As you read, notice the mix of public history and personal prayer. Whether Ezra personally compiled every line or not, the book faithfully presents his ministry and God's restoring work among a scattered people.
And the elders of the Jews builded, and they prospered through the prophesying of Haggai the prophet and Zechariah the son of Iddo. And they builded, and finished it, according to the commandment of the God of Israel, and according to the commandment of Cyrus, and Darius, and Artaxerxes king of Persia.