1-2 - King Cyrus of Persia allows the return to Jerusalem and about 50,000 people go back.
This is about 538 B.C.
3 - They rebuild the altar, and then start on the temple. When they lay its foundation they have a celebration ceremony with singing. The young shout for joy but the old weep, remembering the first temple.
4 - The transplants from Assyria who intermarried with the Israelites left in the land during the exile - generally called Samaritans by the Jews later - ask to build the temple with them. But their theology is off and their worship is syncretistic. See 2 Kings 17:24-41 for the full description and history. Ezra rejects their involvement because of this - not racism but idolatry.
How this is about Jesus
Both the end of Ezekiel and the beginning of Ezra focus on a restored temple.
Jesus claims to BE the new temple (John 2:19-22).
This is a specific fulfillment of prophecy in Isaiah 45, especially verse 13. Assuming Isaiah wrote in the late 700's B.C., the exiles would have known of this prophecy, and of Cyrus by name. So when he came to power and decreed their freedom, they rightly saw God's hand in it.
The whole point of being in the land is to worship God rightly. Modern Jews have mostly forgotten this, it seems. Do not claim the benefits of a covenant relationship with God without being loyal to Him.
When you try to live for God, opposition will arise.