The Old Testament is a book that talks about things that happened a long time ago. Some people in the Old Testament made predictions, or prophecies, about a very special person who would come in the future. This special person is called the Messiah.
The New Testament is another book that talks about the same special person, the Messiah, but this time people are talking about him because he has already come. The people who wrote the New Testament wanted to show that the prophecies made in the Old Testament had come true. So, they quoted the prophecies in the New Testament to show that the Messiah was the person they were talking about.
For example, in the Old Testament, there is a prophecy that says: “Therefore the Lord himself will give you a sign. Behold, the virgin shall conceive and bear a son, and shall call his name Immanuel” (Isaiah 7:14). This means that the Messiah will be born of a woman who has never had a baby before. In the New Testament, this prophecy is quoted when an angel tells Mary she will have a baby, even though she is a virgin. Mary's baby is Jesus, who is the Messiah.
Another example is the prophecy that the Messiah will be born in Bethlehem (Micah 5:2). In the New Testament, when the wise men come to King Herod asking where they can find the Messiah, Herod asks his advisors where the Messiah will be born. They tell him it will be in Bethlehem, just like the prophecy said.
So, when the New Testament quotes the prophecies from the Old Testament, it is to show that the Messiah they were talking about has now come, and that Jesus is that Messiah.