The meaning of Matthew 15:21-28 in the Bible is that Jesus is able to spiritually provide for both the people of Israel and the Gentiles. This is reinforced by a hungry Canaanite woman whose unwavering faith in Jesus causes Jesus to have more mercy on her than his disciples recommend.
The Canaanite woman serves as an illustration of faith, and her interaction with Jesus helps symbolize that there need be no separation between Israelites and Gentiles. Her being a Canaanite is important because the Canaanites had been enemies of Israel in the Bible, whose pagan practices often led Israel into worship of idols.
The woman's refusal to be rebuffed by the disciples and initially by Jesus help showcase how those with nowhere else to turn to should turn to Jesus. Jesus and the disciples at first make it clear that Jesus has a mission to take care of Israel first. She is willing to acknowledge that Israel is more important, but her open acknowledgement of Jesus as the Son of David leads Jesus to proclaim that while he has brought enough bread for Israel, he should have plenty of bread left over for everyone else. This alludes to the upcoming act of Jesus feeding 4,000 people, but also reinforces that both Gentile and Jew can be spiritually saved by Jesus.