Jesus in the Synagogue of Nazareth

After Jesus had been tempted in the wilderness, see The Temptations of Jesus, he returned to Galilee in the power of the Holy Spirit, and the news about him spread throughout the countryside. He taught in their synagogues, and all praised him. He went to Nazareth, where he had grown up, and on the Sabbath day he entered the synagogue, as was his custom. 
He got up to read. (Luk 4:14–16). This occurred
during a visit to his hometown.

According to tradition, seven of those present in the synagogue read from the Tora (Pentateuch or the five books of Moses), that is, the five books: Genesis, Exodus, Leviticus, Numbers, and Deuteronomy. They stood up and read at least three verses each based on a Bible study plan that read the entire Law over the course of a year. Then someone was called forward to read from the prophets; often someone who was visiting was asked to do this. The person who read from the prophets then sat in front of the assembled and made some explanations about the text just read.

The scroll of the prophet Isaiah was handed to him. He unrolled it and found the place where it's written: "The Spirit of the Lord is upon me, because he has anointed me to proclaim good news to the poor. He has sent me to proclaim liberty to the captives and recovering of sight to the blind, to set at liberty those who are oppressed, to proclaim the year of the Lord’s favor." (Luk 4:17–19). 
These verses were known by the Jews to be a messianic prophecy:

• "The Spirit of the Lord God is upon me, because the Lord has anointed me to bring good news to the poor; he has sent me to bind up the brokenhearted, to proclaim liberty to the captives, and the opening of the prison to those who are bound; to proclaim the year of the Lord’s favor." (Isaiah 61:1–2a (ESV)).
• "The Spirit of the Lord is upon me, because he has anointed me; he has sent me to preach glad tidings to the poor, to heal the broken in heart, to proclaim liberty to the captives, and recovery of sight to the blind;
  to declare the acceptable year of the Lord." (Isaiah 61:1–2a (LXX)).

Only the Septuagint (LXX) mentions "recovery of sight for the blind", see 
Which Version of the Old Testament did Jesus Use? Jesus spoke Aramaic, 
the language spoken in Israel at that time, and probably also Hebrew, which
was used in the synagogues where it was interpreted into Aramaic. If he read from the Septuagint (LXX), which was written in Greek, he must also have known Greek.
However, it's more likely that this sentence was part of a Proto-Masoretic text in use at the time. That would mean that Jesus was reading from a text in Hebrew.

Jesus read aloud from the prophet Isaiah. The passage in question (Isaiah 61:1–2a) foretells the earthly ministry of Messiah "the anointed one". It talks about:

• Proclaim the good news to the poor, see Blessed are the poor in spirit
  in The Sermon on the Mount. 
• Liberty from addictions, from captivity in sin (Gal 5:1), see Salvation.
• Sight for the blind, see Blind People in Miracles of Jesus and
  Laodicea in The Seven Churches in the Book of Revelation.
• Freedom for the oppressed, those who have become victims, see the last paragraph in Blessed are the poor in spirit in The Sermon on the Mount.
• Proclaiming the year of the Lord’s favor, speaking in a general sense of the day of Salvation. Messiah comes to proclaim the day of God's acceptance.
  To compare with Sabbath year and Jubilee year, see
  The Book of Habakkuk.

Jesus stops reading after the first phrase of Isaiah 61:2. This relates to his interpretation that this part of the prophecy is being fulfilled at that very moment. Isaiah's prophecy also speaks of Messiah coming with God's righteous vengeance (Isaiah 61:2b), but that prophecy will wait until Christ's second coming 
(Rev 19:11–16), see Prophecies of Jesus' Second Coming. 

Then he rolled up the scroll, handed it back to the attendant, and sat down.
All eyes in the synagogue were fixed on him. He began by saying to them:
"Today this scripture is fulfilled in your hearing." Everyone spoke well of him
and were amazed at the gracious words that came from his lips.
But they asked: "Isn’t this Joseph’s son?" (Luk 4:20–22, Mark 6:1–3).

The local people were skeptical to Jesus' claim to be Messiah. They wondered how a simple local whose family they knew could make such a claim. (Mat 13:54–57a).

Jesus said to them: "Surely you will quote this proverb to me: 'Physician, heal yourself!' And you will tell me: 'Do here in your hometown what we have heard that you did in Capernaum.'" He continued: "Truly I tell you, no prophet is accepted in his hometown." (Luk 4:23–24, Mat 13:57b–58, Mark 6:4–6). Jesus then said:

1. "I assure you that there were many widows in Israel in Elijah’s time, when the sky was shut for three and a half years and there was a severe famine throughout the land. Yet Elijah was not sent to any of them, but to a widow in Zarephath in the region of Sidon." (Luk 4:25–26, 1 Kings 17:8–24).
   See Ahab of Israel in Elijah the Prophet.
2. "And there were many in Israel with leprosy in the time of Elisha the prophet, yet not one of them was cleansed—only Naaman the Syrian." 
   (Luk 4:27, 2 Kings 5:1–14). See Jehoram of Israel in Elisha the Prophet.

All the people in the synagogue were furious when they heard this. They got up, drove him out of the city, and brought him to the edge of the hill on which the city was built, to throw him off the cliff. But he went straight through the crowd and continued on his way. (Luk 4:28–30). It was a miracle that Jesus could do that.

At first the audience received Jesus' teaching with optimism. But when Jesus spoke of God giving grace to the Gentiles, they became violent. Most Jews at the time of Jesus were raised with the belief that God only loves the Jews and that the Gentiles weren't part of the salvation, however see Peter Sent to the Gentiles.

See also Some stories in the gospels at the end of The Four Gospels.