What did Jesus mean when He said don't give children's food to dogs?

One of the verses to which you refer is Matthew 15:26But he answered and said, It is not meet to take the children's bread, and to cast it to dogs.
See All... - "But he answered and said, It is not meet to take the children's bread, and to cast it to dogs." In this story (Matthew 15:21-28 [21] Then Jesus went thence, and departed into the coasts of Tyre and Sidon. [22] And, behold, a woman of Canaan came out of the same coasts, and cried unto him, saying, Have mercy on me, O Lord, thou son of David; my daughter is grievously vexed with a devil. [23] But he answered her not a word. And his disciples came and besought him, saying, Send her away; for she crieth after us. [24] But he answered and said, I am not sent but unto the lost sheep of the house of Israel. [25] Then came she and worshipped him, saying, Lord, help me. [26] But he answered and said, It is not meet to take the children's bread, and to cast it to dogs. [27] And she said, Truth, Lord: yet the dogs eat of the crumbs which fall from their masters' table. [28] Then Jesus answered and said unto her, O woman, great is thy faith: be it unto thee even as thou wilt. And her daughter was made whole from that very hour.
See All...), a woman of Canaan came to Jesus to ask Him to deliver her daughter from a devil. Initially, He refused with the statement: "I am not sent but unto the lost sheep of the house of Israel" (v.24).

When she persisted and refused to give up, Jesus made the statement in question about giving the bread of the children to dogs. The woman still would not turn away. She pointed out that even the dogs get to eat the crumbs that fell from the table. At this, Christ declared the greatness of her faith and delivered her daughter.

In this story, Jesus is showing us that His main calling during His lifetime was to the Jewish people and not to other peoples of the world. When He sent out the twelve disciples in Matthew 10, He told them, "Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not: But go rather to the lost sheep of the house of Israel" (v.5-6). Even Paul testified "that Jesus Christ was a minister of the circumcision" (Romans 15:8Now I say that Jesus Christ was a minister of the circumcision for the truth of God, to confirm the promises made unto the fathers:
See All...). When the Bible refers to the "circumcision," it is referring to the Jewish people.

You see, Jesus was born as the Jewish Messiah--as their promised Holy One. He came to them to fulfill all the Old Testament prophecies concerning Him. It was only after He was rejected by them, that the message was offered to all people.

So, the children of Matthew 15:26But he answered and said, It is not meet to take the children's bread, and to cast it to dogs.
See All... are the Jewish people. The dogs are a picture of the Gentiles--that is, everyone who is not a Jew. Jesus came to His own people (John 1:11He came unto his own, and his own received him not.
See All...). Just as it would not be right to allow our children to go hungry while we fed our pet dogs very well, Jesus would not spend His ministry reaching the Gentiles when He was called to go to the Jews.

The good news is that Jesus was willing to give crumbs to the woman of Canaan because of her faith even when His ministry was to the Jews. And now, after the rejection of Jesus by the Jewish people, "salvation is come unto the Gentiles" (Romans 11:11I say then, Have they stumbled that they should fall? God forbid: but rather through their fall salvation is come unto the Gentiles, for to provoke them to jealousy.
See All...). According to the apostle Paul, we Gentiles were "without Christ, being aliens from the commonwealth of Israel, and strangers from the covenants of promise, having no hope, and without God in the world" (Ephesians 2:12That at that time ye were without Christ, being aliens from the commonwealth of Israel, and strangers from the covenants of promise, having no hope, and without God in the world:
See All...). Yet through Jesus Christ we "who sometimes were far off are made nigh by the blood of Christ" (Ephesians 2:13But now in Christ Jesus ye who sometimes were far off are made nigh by the blood of Christ.
See All...). We have been given the full blessings of being the children of God. And, though Jesus will again reach out to the Jewish people, we know that He will never turn from us because we are Gentiles.