The passage that Jesus reads from the book of Isaiah applies to Jesus’s work because it sets out rather clearly the idea that Jesus has come to reach out to those who are not the elites of society. He has come to include the poor and oppressed in the Kingdom of God and he has come to minister to those who are looked down upon by society. This would have been very important to Luke as he was the only Gospel writer who was a Gentile.
In his ministry, we know that Jesus constantly ministered to those who were looked down upon by the religious leaders of his society. He was willing to touch lepers. He was willing to forgive adulterers and prostitutes. He ate with tax collectors. One of his last acts as a human being was to speak to a common criminal and to tell him he would enter the Kingdom of Heaven. By reading the passage that he did from Isaiah, Jesus is emphasizing this aspect of his ministry. He is proclaiming that he has not come for the righteous but for the “sick” who need a “doctor.”
Luke would have been very receptive to this message as it shows Jesus rejecting the idea that the Jews were somehow more favored. Instead, it shows Jesus coming to announce a new way of honoring God that would be more inclusive.