10 And as Jesus reclined at table in the house, behold, many tax collectors and sinners came and were reclining with Jesus and his disciples.
11 And when the Pharisees saw this, they said to his disciples, “Why does your teacher eat with tax collectors and sinners?”
12 But when he heard it, he said, “Those who are well have no need of a physician, but those who are sick.
13 Go and learn what this means, ‘I desire mercy, and not sacrifice.’ For I came not to call the righteous, but sinners.”
The allusion to Hos. 6:6 LXX (‘I desire mercy and not sacrifice’) brings into view the wider context of Hosea’s prophecy. The people of Israel will take their sacrificial animals to the temple, but they will not find the Lord there (Hos. 5:6). He has withdrawn from them until they acknowledge their guilt and seek his face, saying: ‘Come, let us return to the Lord; for he has torn us, that he may heal us; he has struck us down, and he will bind us up. After two days he will revive us; on the third day he will raise us up, that we may live before him’ (Hos. 6:1-2).
Here is the significance of the saying about the physician. Israel is sick (figuratively) because it is under judgment; if the people repent and turn to the Lord, they will be healed - and significantly will be raised up on the third day to live before God. The tax collectors and sinners perhaps then represent that part of the population that turns to the Lord in their distress looking for healing and resurrection.
The healings in the Gospels are, of course, a concrete prophetic enactment of this forgiveness.