Summary: Luke dives right in: he exclaims that this is all true, from Zechariah and Elizabeth's miraculous birth of John to visits by an angel of the Lord to both Zechariah and Mary, Elizabeth's cousin, about two miraculous births.
Response: This is a classic my-family way to start a story: completely unrelated and extremely long-winded. But the Greek is very insistent, first with "it happened that" (no really, this happened!) and the Greek "Kai" (then this happened), to keep the story going and going. God didn't pop in one day and die on the cross the next weekend. Like any life, Jesus' life is its own story mixed in with that of all around Him.