So has the Lord done for me at a time when he has seen fit to take away my disgrace before others. The conception and birth of John the Baptist are somewhat similar to those of his cousin Jesus which they introduce. It was not a random choice that God let Zechariah and Elizabeth to be the parents of Jesus’ forerunner. God was making a statement. And it was a statement about what the birth and ministry of Jesus was going to be, something that Elizabeth sums up in her words quoted above.
Up until the conception of John the Baptist, Elizabeth was considered barren. In the eyes of other women, she was a cursed woman. She (or her family) must have done something that made God to mete out such a punishment on her. Upon their deaths (she and her husband), they would have no one to ‘carry their names’ through to succeeding generations. Theirs would be a real death for they would have left nothing behind with which they could be remembered. It was a  misfortuneabout which a barren woman would be reminded on a daily basis. One can thus understand the joy with which Elizabeth greeted her coming to know of the good news that she was expecting a child (no wonder she went into exclusion for five months!)
Elizabeth’s joy prefigured the joy that was going to surround and accompany the birth of Jesus Christ. In the words of Zechariah her husband, the birth of Jesus was going to mark a new dawn for countless men and women (cf. Luke 1:78). Like Elizabeth, there were many who had no future, many who were reminded on a daily basis of their misery, many who were considered non-persons and as such were unworthy of God. In Jesus Christ, these sons and daughters of God were going to find a friend and someone who cared about their wellbeing. It was on behalf of them that Elizabeth spoke those words of praise and thanksgiving: in Jesus Christ, God was going to take away their shame and disgrace.