These vinyl or leather straps are usually sewn on for one of three reasons. One, they are stitched onto the reverse side of the rug to prevent "curling edges." Curling edges most often occurs when the rug is hanging in a showroom as the natural tendency for a rug under it's own weight is to curl at the bottom. Secondly, these straps may be applied to fix another form of curling edge resulting from too much internal tension of the rug more commonly known as buckling. Third, and finally, these straps are applied to increase the weight of the rug so if the rug is kicked up, it will fall back in place rather than stay folded upon itself.
If you prefer to remove the straps you may, however, it is better to leave them on indefinitely as one may not know whether they were applied as a preventative measure or (less likely) to fortify the rug's structure.
As you can see in the photograph above, some carpets are outfitted with this binding on either edge of the rug.
The green arrow above shows where an appropriate area would be to snip if you wish to remove the strap. While we do not suggest removing the strap as it has probably been sewn on for a reason. If you carefully look at the way your rug lays over several weeks, you may be able to decide whether or not the strap is necessary. Allow the rug to settle for some time prior to making this decision.