A "run" in fabric is where a thread has been pulled and leaves a gap in the weave. sometimes you will see a loop of thread which often can ne eased back gently along the fabric, but if the thread is no longer there , then , it's very difficult to fix it, and I suggest that you have to accept it. it looks like it has been fringed on the sides, and perhaps one of those threads got pulled .
I wouldn't hem the other sides, it will spoil the drape , and also make it narrower and also need a double (folded zig-zag)sort of hem if you wanted to keep the fringe.
The Great kilt was an everyday garment, and not anything special in many ways so I would except to see a few runs or even tears as part of normal use- sorry that's probabaly not what you wanted to hear.