57 Now the time came for Elizabeth to give birth, and she bore a son. 58 And her neighbors and relatives heard that the Lord had shown great mercy to her, and they rejoiced with her. 59 And on the eighth day they came to circumcise the child. And they would have called him Zechariah after his father, 60 but his mother answered, “No; he shall be called John.” 61 And they said to her, “None of your relatives is called by this name.” 62 And they made signs to his father, inquiring what he wanted him to be called. 63 And he asked for a writing tablet and wrote, “His name is John.” And they all wondered. 64 And immediately his mouth was opened and his tongue loosed, and he spoke, blessing God. 65 And fear came on all their neighbors. And all these things were talked about through all the hill country of Judea, 66 and all who heard them laid them up in their hearts, saying, “What then will this child be?” For the hand of the Lord was with him.

67 And his father Zechariah was filled with the Holy Spirit and prophesied, saying, 68 “Blessed be the Lord God of Israel, for he has visited and redeemed his people 69 and has raised up a horn of salvation for us in the house of his servant David, 70 as he spoke by the mouth of his holy prophets from of old, 71 that we should be saved from our enemies and from the hand of all who hate us; 72 to show the mercy promised to our fathers and to remember his holy covenant, 73 the oath that he swore to our father Abraham, to grant us 74 that we, being delivered from the hand of our enemies, might serve him without fear, 75 in holiness and righteousness before him all our days. 76 And you, child, will be called the prophet of the Most High; for you will go before the Lord to prepare his ways, 77 to give knowledge of salvation to his people in the forgiveness of their sins, 78 because of the tender mercy of our God, whereby the sunrise shall visit us from on high 79 to give light to those who sit in darkness and in the shadow of death, to guide our feet into the way of peace.” 80 And the child grew and became strong in spirit, and he was in the wilderness until the day of his public appearance to Israel.

Luke is the only gospel of the four gospels (Matthew, Mark, Luke, John) that gives more specific attention to Zechariah. Both Zechariah and Elizabeth were from the tribe of Levi, and Zechariah was a priest (Luke 1:5). The Old Testament priestly duties are somewhat complex (read more here) but certainly included preparing sacrifices on behalf of the people, guarding the temple, cleansing and purifying, and conveying God's word to the people of Israel. 

Zechariah's priestly prophecy is significant in that A) he makes the connection that his unborn son's role is to be a prophet that preaches the coming King (Luke 1:76), B) he establishes the coming King to the Old Testament prophets (Luke 1:69-70,72-73), and C) he glimpses and exults in the great gospel and good news of God-- that the coming King is to bring forgiveness of sins, mercy, light into a dark world, and peace to sinners (Luke 1:77-79). 

Luke is likely writing the book of Luke to Gentile Christians and may have added this part to teach and emphasize Jesus Christ's fulfillment of Old Testament prophecy and law.