Several accounts of the things fulfilled have been written, and Luke wanted to write, too, having thorough knowledge of it all.
Zachariah, a priest married to Elizabeth but with no children, offers incense and the angel Gabriel appears and promises him a son to name John. He will be a Nazirite, the Elijah coming before the Messiah (Malachi 4:5-6). Zachariah doesn't believe, and so will be mute until he is born - 9 months!
Elizabeth conceives and rejoices. Gabriel goes to Mary 6 months into Elizabeth's pregnancy. He announces the birth of Jesus, and that He will be the eternal King of Israel. Mary asks the same question Zachariah asked, but from faith, not doubt. The Holy Spirit will mysteriously bring about her conception. Mary goes to visit Elizabeth. Elizabeth blesses Mary, led by the Spirit, and calls her the "mother of my Lord."
Mary sings a song of Scripture (Magnificat), praising God for scattering the proud and filling the hungry, for helping Israel as He told Abraham He would.
John is born, Zachariah assets to name him John, and can immediately speak. This is talked all over. Zachariah prophesies, blessing God for redeeming Israel as He promised Abraham He would. He tells baby John that he will go before God's coming as a messenger. God is bringing knowledge and light by forgiving their sins.
We bless God for keeping His Word.
We take Him at His Word, not doubting that word.
We know His Word well enough to know when and how He kept it, and quote it in songs (which Mary and Zachariah do here).