This prophecy is spoken by Zechariah, the father of John the Baptist. It is spoken on the occasion of John’s ritual circumcision. Zechariah had been struck dumb by God for a lack of faith, but at this point, he shows that he has true faith in God and is allowed to speak again. His prophecy essentially has two parts.
In the first part of the prophecy, (verses 68-75), Zechariah is prophesying about Jesus. He is saying that God is sending someone from the House of David who will rescue the Jews from all of their problems. He is saying that God has finally (as he promised) sending a figure to save the Jews from oppression.
In the second part of the prophecy, Zechariah turns to prophesying about his own son. Having talked about Jesus, Zechariah is now talking about the role that John will play. He is prophesying that John will prepare the way for Jesus. John will act as a herald for what Jesus is going to do. This will be borne out as John becomes a prophet and essentially starts Jesus’s ministry by publicly baptizing him.