" Bright girdle" represents old religious ideas. Matthew Arnold sounds melancholic; he would like religious ideas to stay. He is complaining against the advancement of rational and industial times. Therefore, the word "girdle" may represent the " Sea of Faith" because , in the past, faith was unquestionable. The folds of a belt ( girdle) were kept "furled" , that means fastened. Faith was accepted and rigid like a bright girdle / bright ribbon or belt.
It refers to the feeling of support and belonging that people once felt in happier times.
