How to fix frayed carpet at tile transition
I installed carpet with a professional for two years right out of college.  I don't think you need to buy anything.  You need to pry the lip of that transition piece up so you can get the carpet loose along the entire length, then stretch the carpet forward more with a knee kicker.  You can hold the carpet in place with the kicker while you use a rubber mallet to hammer down and lock the carpet under the transition again.  If the carpet is frayed bad, you can trim a little to give the tabs more meat to grab onto.  Most guys could do this in 5 minutes.  If you call a local installer and ask him to swing by one day when they are in the area, they may do it for $50 cash.