Use a damp washcloth to clean the damaged areas of the rug. Scrub the areas thoroughly, but don't be too aggressive as to fray or further damage the leather. Cut off the excess fabric fringes, and loose fabric tears with a pair of scissors. Cut the frays and loose fabric as close to the rug as possible.
Purchase a leather repair kit from your local hardware or craft store. Apply some leather filler to each damaged area on the rug. Allow each leather filler application to dry for about 10 to 15 minutes, and add another layer of filler. Repeat this until each damaged area is filled completely with leather filler.
Use a putty knife to wipe a thin layer of leather filler over the treated areas, and allow it dry for 15 minutes.