Seems like an odd thing to celebrate, Mary high-tailing it out of town to hang out with her cousin...suspiciously like the old story of the unmarried pregnant girl shipped off to live with the older spinster/widowed aunt until after the birth.
But that is certainly not the case. She has gone to aid her cousin in her own time of need. Elizabeth has probably aided many others in her day but this is a first for her - she is expecting in her later years.
But the most important thing we celebrate is once again the showing of Jesus to the world as will happen at his birth, his day in the temple, and many others until he his lifted highest on the Cross and then into Heaven. This feast is less about Mary visiting Elizabeth and more about John recognizing Jesus in the womb of the Theotokos. Jesus' divinity shines forth even in his developing humanity. Elizabeth recognizes the irony of the visit: "And how does this happen to me, that the mother of my Lord should come to me?" (Luke 1:43) Who am I to be served by you? This is echoed at the Last Supper by Peter, and like Mary's response, Jesus calls him to remember the purpose for it all.
This is our day to "leap for joy" in the womb of Mother Church and proclaim the glory of God to all the Earth.
Mary set out and traveled to the hill country in haste to a town of Judah, where she entered the house of Zechariah and greeted Elizabeth.
When Elizabeth heard Mary's greeting, the infant leaped in her womb, and Elizabeth, filled with the Holy Spirit, cried out in a loud voice and said,
"Most blessed are you among women, and blessed is the fruit of your womb.
And how does this happen to me, that the mother of my Lord should come to me?
For at the moment the sound of your greeting reached my ears, the infant in my womb leaped for joy.
Blessed are you who believed that what was spoken to you by the Lord would be fulfilled."
And Mary said:
"My soul proclaims the greatness of the Lord;
my spirit rejoices in God my Savior,
for he has looked with favor on his lowly servant.
From this day all generations will call me blessed:
the Almighty has done great things for me,
and holy is his Name.
He has mercy on those who fear him
in every generation.
He has shown the strength of his arm,
he has scattered the proud in their conceit.
He has cast down the mighty from their thrones,
and has lifted up the lowly.
He has filled the hungry with good things,
and the rich he has sent away empty.
He has come to the help of his servant Israel
for he has remembered his promise of mercy,
the promise he made to our fathers,
to Abraham and his children for ever."
-- Luke 1:39-56