Wine played a vital role at wedding feasts in Jesus' time, marking the beginning of a marriage. This is evident from the story of the Wedding Feast at Cana, which precedes today's Gospel reading. When Jesus referred to Himself as the bridegroom, He compared His presence in the world to that of new wine.
The new wine, representing the Lord's loving presence and life-giving action, requires new wineskins to contain and share it. While the Lord's love is a gift, it also places certain demands on us, urging us to continually renew our lives so His love can truly shape us.
With new wine comes the need for new wineskins: we must continually shed our old ways and develop anew, thus always growing closer to God.