If one looks in the Me-asef Kol hamachanos on O.C. 21 he says that depending on how one learns the Gemorra in Brochos it is either Ossur or Muttar to drag the tzitzis on the ground. There are explanations that the Gemmorah was talking about tzitzis dragging on the ground of a cemetary where it would be Ossur because of Loeg L’rosh. (Teasing the dead who can no longer do the Mitzva) There are others that learn the Gemorah Simply that dragging Tzitzis on the ground ANYWHERE is ossur!
The MB learned the second way, hence he says to put the tzitzis in one’s Gartel or belt.