Theophylact of Bulgaria
The wife demonstrated a passionate faith when she learned about Him. Consequently, the Lord chose not to fulfill her request right away but rather postponed the blessing to illustrate her steadfastness and her patient waiting despite the initial denial. We should also take this as a lesson not to abandon our prayers at the first hint of silence, but to persist in our petitions until we obtain what we seek. The Lord refers to the Gentiles as ‘dogs’ due to their being deemed impure by the Jews. The ‘bread’ symbolizes the grace that God has designated for His "children," meaning the Jewish people. Thus, He indicates that the Gentiles should not receive the grace reserved for the Jews. The wife’s wise and faithful response led her to receive the blessing she sought.
As fallen individuals, we too are labeled ‘dogs,’ filled with impurity, and therefore we are unworthy to receive the ‘bread’ of God; that is, to partake in the holy Mysteries.