Shaving does not make hair thicker. It only makes your experience of the hair seem thicker. This illusion comes about because a "virgin" hair tapers to a soft point, and has lots of give when you brush across it. A shaved hair comes to either a blunt end, or a sharp point with a shiny spot where it was sliced by the blade. This does a few things. It makes the hair more bristley to the touch, and catches light with the same kind of sparkle seen on a "disco ball" when the sun hits the "slice point" just right. Also, since the shaft of the hair is in fact thicker than the point of a virgin hair, it does seem to be thicker if allowed to grow to stubble, because it doesn't taper off at the ends anymore.
Rest assured, however, shaving is perfectly safe. No one grows thicker, deeper, or even more plentiful hair from simply shaving.