It's when you don't feel comfortable because of some kind of anxiety that grows. The word "tugging" is used since it's sort of doing that to you towards unease. You could also say "a nagging sense of unease" which is the same thing, when a feeling of uneasiness nags (pesters) you. Hopefully at least one of these will make more sense to you!

Tug means to pull usually, but in this case, it means that feeling you have that something is wrong.