Need clarification on Matthew 15:21-28

It was indeed the faith and humility from a Canaanite woman who was an outcast for the Jews. She beg and insisted that if she could only partake even leftovers (which is not intended for them either) from the Jews just for the healing of his daughter who was demon-possessed then. She exampled herself as a dog (as Jews treat them as is) which indeed had the privilege to eat his master's leftovers. This woman definitely heard what Jesus did to others and knowing Him to be very compassionate, she claimed to herself that healing will also be possible to her daughter (even if it beyond the Jewish norms). It's her faith to Him that made her request be granted.

For this, you have to understand the relationship the Jews had with the Samaritans, which the woman at the well was. Samaritans were the people the Jews referred to as those who had been exiled, of the northern tribes, by the Assyrians. They had inter-married with those around them. So, Jews viewed them as "half-breeds." Sometimes they would have even referred to them as dogs. Jesus, then, used this terminology, not as a truly racist comment, as would have been the norm, but to provoke her to dig deeper. This she did by making the comment she did, as the Samaritans fully understood the Jews feelings and terminology. Hope that sheds some light on the subject.

Ah yes, they were right - the woman was a Gentile, and the Jews look down on the Gentiles because they're not God's chosen people. Jesus used the parable to say that salvation can also come to the Gentiles. God isn't just the God of the Jews, after all.