It's a reenactment of the biblical story of Mary and Joseph searching for lodging in Bethlehem on the night Jesus was born.
In the tradition, groups of carolers go from house to house, singing carols and asking for permission to enter. The homeowner represents an innkeeper, and they usually refuse the couple entry, but eventually, they are allowed in.
So, in English, you could call it "The Search for Shelter" or "The Christmas Caroling Tradition" or even "The Filipino Christmas Play."