It will NOT become thicker; that is an urban legend. It Will, however, be "scratchy" as the razor cuts the hair at an angle, leaving a sort f microscopic 'blade' tip, which makes it feel rougher than it did before.
Absolutely nothing, and shaving is even recommended. Considering the ridiculous $15 price tag on Olay's new facial hair removal rip off, go for it. It's just another marketing trick, another product you don't need. For instance, for years, I have used an expensive car wash that had a"special" wax in it. Ran out last week, used Dawn liquid, and my car has never been shinier!