In Jesus' day, it was common for many of the Jewish people to hold themselves aloof from others who were not the descendants of Abraham. They did not want to get themselves in a situation in which their religion would be compromised or in which others would think that they saw all religions as the same. They even went so far as to call the Gentiles "dogs" because they were ritually impure and could not offer sacrifices in the Temple. Yet throughout the Old Testament, it was promised that God would bless all people through the descendants of Abraham. Furthermore, God's kingdom was to be opened up for all peoples.
In the New Covenant, however, the relationship was not going to be based on bloodlines or family ties, but rather faith in Jesus. (Note that Jesus said Blessed are those who hear the word of God and observe it Luke 11:28). So Jesus tested the woman who desired the healing of her daughter. He said It is not right to take the food of the children and throw it to the dogs (Matthew 15:26). Her response was Please, Lord, for even the dogs eat the scraps that fall from the table of their masters (Matthew 15:27) brought forth our Lord’s approval. One might say that she humbled herself before the Lord. After all, before the Almighty God, who is clean and worthy of anything? She might have acknowledged herself as lowly, which is pleasing to the Lord. Her statement revealed that she trusted that Jesus had the power to heal her daughter. Just like the animals can trust in the benevolence of good masters, so could anyone trust in the goodness of the Lord. His blessings and His love was always overflowing to others. Because she believed in Him and trusted in His goodness even in the face of seeming rejection, Jesus approved of her and healed her daughter.
Many times we pray for things- maybe a privilege, a healing, advice or help, and the answer does not seem to come. Or maybe it appears that the answer we get is negative and God is just not very interested in our plight. We should not give up. If we continue to cry out to God, He will help. Of course, sometimes the answer is not what we wanted, maybe it is "no." But it is always "because I love you and want to give you something better." With God we are not animals, we are His children.
Jesus seeks faith and trust.