This is a very good question, Darlene! One which many should be asking, rather than avoiding. Many of the answers above or typical "Christian" responses; however, they are not completely accurate.
First, there is a difference between a New Testament "Jew" and the lost sheep of the house of Israel. If you have read all the way through Matthew, you probably picked up on some conflicting statements from Jesus regarding these groups and his purpose vs. what people currently teach that his purpose was.
The "lost sheep of the house of Israel" did not kill, nor did they reject Jesus.... They were the ones following him and John the Baptist throughout the land. Remember that Jesus was purposed to lead his people and he did so! He was never sent here to change or save the Pharisees who are often equated to the "Israelites" by the generic term "Jew".
Then went out to him Jerusalem, and all Judaea, and all the
region round about Jordan, And were baptized of him in
Jordan, confessing their sins. Mat 3:5-6
And there followed him great multitudes of people from Galilee, and
from Decapolis, and from Jerusalem, and from Judaea, and from beyond
Jordan. - Mat 4:25
John makes the distinction between the "Jews" who include the Pharisees who were religious and political leaders and the people who were Israelite descendants and considered the lost sheep of the house of Israel. The Pharisees who actually had Jesus killed are never said to be descendants of Jacob or Judah, therefore they are never said to be strictly Israelites.
Mat 3:7 But when he saw many of the
Pharisees and Sadducees come to his baptism, he said unto them, O
generation of vipers, who hath warned you to flee from the wrath to
come? Mat 3:8 Bring forth, therefore, fruits meet for repentance: Mat
3:9 And think not to say within yourselves, We have Abraham to our
father: for I say unto you, that God is able of these stones to raise
up children unto Abraham.
From John's encounter with the Pharisees, it is immediately clear the Pharisees and Sadducees were never part of the redemption and salvation plan. They were identified as the enemy, the serpent, who would, according to John, burn in hell. This is made clear right off the bat in the New Testament.
It was descendants of Jacob specifically, and not of Abraham, who was given the promise of a Messiah. Which is why John made the comment that God could make rocks descendents of Abraham. Remember that Abraham had many descendents who became great nations including Esau, Ishmael, and the sons of Keturah, his second wife. The descendants of Jacob are the lost sheep of the family of Israelites, who were without proper leadership whom Jesus was sent for.
So the short answer to your question is that Jesus says that he was only sent for these certain people because he was. Afterall, Israel has a special mission and Jesus came to help them carry that mission out, which is why he and his disciples only taught the lost sheep of the house of Israel. [Mat 10:6]
However, he also demonstrates through the story of the Canaanite women that from times of old, people, also called sojourners and strangers in the old testament, had always been grafted into the family and became a part of the family and the same still holds true. While they may not have the same mission as Israel, they too can receive the benefits through their faith in the God of Israel. Also, according to scripture, these adopted children must also submit to the complete law or Word of the God of Israel to be grafted in... Im not talking about the ten commandments, but the WHOLE message and purpose of the God of Israel.
Also the woman says some things that should not be ignored.
Then came she and worshipped him, saying, Lord, help me. -Mat 15:25
So she knew who he was and that he was someone to be bowed to. But then Jesus essentially calls her a dog.
And she said, Truth, Lord: yet the dogs eat of the crumbs which fall from their masters' table. -Mat 15:25
She agrees that she is a dog!
First, this takes us back to a great "Israelite" who was a fearless leader but was not actually an Israelite. Rather, he was either an Edomite or Canaanite. He was a decadent of Kenaz, son of Esau [Gen 36:11], who lived in the Canaanite's land [Gen 15:19] and were therefore also a Canaanitish people. Also, Esau married Canaanite women, making his descendents Cannanitish! His name was Caleb, which means "dog" in Hebrew.
Save Caleb the son of Jephunneh the Kenezite, and Joshua the son of Nun: for they have wholly followed the LORD. -Num 32:12
So the moral of the story is that Jesus, whose name was really Yahushua, was sent to his own people as promised in Old Testament scripture, but since the beginning of Israelite history, when people submitted to the God of Israel they became like Israel in so many ways. They never become Israel, but like the Canaanite women point out, others are allowed to eat from their table. Which means they share the same benefits as the Israelites when they make themselves as one, having been adopted. The Canaanite women knew scripture and reminded Jesus of this truth. She held him to the word of God which demonstrated her faith in the truth of God's word.
I hope this helps to give you a complete understanding of this statement and other's like it that Yahushua made!