Isaiah 61 is the anointed one's self-description: 'The spirit of the Lord is upon me, because the Lord hath ANOINTED ME: he hath sent me to preach to the meek, to heal the contrite of heart, and to preach a release to the captives... to proclaim the acceptable year of the Lord' (Isaiah 61:1–2). 'Anointed' is the word itself — Messiah in Hebrew, Christ in Greek.
Luke places the scene at the front of the public ministry: in the Nazareth synagogue on the Sabbath, Jesus stood up to read, was handed the scroll of Isaiah, found this passage, read it, closed the book, sat down — 'and the eyes of all in the synagogue were fixed on him. And he began to say to them: THIS DAY IS FULFILLED this scripture in your ears' (Luke 4:16–21). It is the only place in the Gospels where Christ reads a prophecy aloud and claims its fulfillment in the first person, in the same breath.
The sequel belongs to the record too: His own townsmen, offended, drove Him out and tried to throw Him from the hill's brow (Luke 4:28–30) — the first rehearsal of the rejection the prophets had also foreseen. He noted it Himself: 'No prophet is accepted in his own country.'