The angel is actually quoting or paraphrasing Scripture to her, particularly two Messianic prophecies from Isaiah that many religious Jews probably had on the tips of their tongues. They were expecting Messiah to come soon, and knew these prophecies had to come to pass for Messiah to be born.
The first is from Isaiah 7:14: "Therefore the LORD Himself will give you a sign: Behold, the virgin shall conceive and bear a son, and shall call His name Immanuel." Immanuel means "God with us." Gabriel inserts a different name, one that God's Son would normally be called: Jesus, which means "Savior." It is really not so different since only God Himself can save.
The second part of Gabriel's paraphrase comes from Isaiah 9:6-7:
For unto us a Child is born, unto us a Son is given; and the government will be upon His shoulder. And His name will be called Wonderful, Counselor, Mighty God, Everlasting Father, Prince of peace. Of the increase of His government and peace there will be no end. Upon the throne of David and over His kingdom, to order it and establish it with judgment and justice from that time forward, even forever. The zeal of the LORD of hosts will perform this.
How did the angel convince Mary of what was happening? He quoted Old Testament prophecies to her! In effect, he tells her, "Look, Mary. God has chosen you to fulfill these prophecies."
In response, she asks a very practical question: "How can this be? I can't have a baby. Joseph and I have not consummated the marriage." He replies to her in a parallelism, a form of speech that Hebrew and Aramaic speakers often used to add detail to their statements: "The Holy Spirit will come upon you," and then he defines what he means: "And the power of the Highest will overshadow you." Putting these two clauses together, he defines the Holy Spirit as the power of the Highest; it is God's ability to effect this miracle.
The angel's use of "overshadow" was undoubtedly comforting to her. To us, it might sound intimidating to be overshadowed by the power of the Highest, but Mary, well-versed in Scripture, gives no reaction that it frightened her. Perhaps she thought of Exodus 40:34-38, in which similar language is used of God covering the Tabernacle in the wilderness with the pillar of cloud and fire. To an Israelite, it was comforting to think that God would hover above them like an eagle over its nest, with wings outspread, protecting, providing, and helping.
It may have also made her think of the constant miracles that God did on behalf of His people in the wilderness. God provided for them constantly for forty years, and the Bible is clear that nothing happened unless God allowed it. Through Gabriel, God was telling Mary, "I'm going to take care of all of this. There is no need to worry." And apparently, her anxieties disappeared.
God then gives her a sign to confirm what He has just said. He tells her to visit her cousin, Elizabeth—an old, barren woman, whom she would find to be six months pregnant! This was also a sign to show Mary that everything would be fine. When she went to see her cousin (Luke 1:39-42), the as-yet-unborn John the Baptist leaped in Elizabeth's womb, confirming to both Elizabeth and Mary that everything that they had heard was true. Moreover, Elizabeth repeats what the angel said to Mary: "Blessed are you among women. Blessed is the fruit of your womb" (verse 42).
Verse 37, "For with God nothing will be impossible," is another comforting reference to the Old Testament. A more literal translation of his statement would be, "For no saying from God shall be void of power," or "For no word from God shall be powerless." This makes it a paraphrase of Isaiah 55:11: "So shall My word be that goes forth from My mouth; it shall not return to Me void, but it shall accomplish what I please, and it shall prosper in the thing for which I sent it."
In effect, he assures her, "This is certain because God has said so." Her response reflects that she is completely convinced by this: "Behold the maidservant of the Lord! Let it be to me according to your word" (Luke 1:38). This is reminiscent of Hannah's attitude in I Samuel 2. Like her, Mary submits unconditionally to God's election of her for this task. She says, "I am the Lord's servant. He can do with me what He will." She gives her life to it.