This is a fix I made on a "medium grade" rug, made with rows of stitching. Nicer plush rugs may not work for this.
It also helps if the rug is a little multicolored to help hide your repairs.
I'll go over a small and large snag, with photos. I already just about finished the large one when I decided to document it.
Flexible fabric glue (I opted for some that dries clear and flexible)
Straight pins (or something similar, for the large pulls that yield long pieces of string/yarn/fiber)
Scotch tape (as needed)
Something heavy like a text book
Step 1: Trim
First, cut some of the snag away. Just be sure you leave enough to cover the void that it created. Many times the snags fluff and fray and will never be the same shape they were originally.