Before I explain myself, let's briefly define what people mean when they say "thicker"; I usually hear two different ideas. 1 is that there is more hair growing back, and the other is that the hair follicle itself is thicker.
It seems that most people are told this saying when they are still in puberty, when all sorts of changes are happening to your body, including the growth of new hair. If a pubescent boy shaves his beard, more hair seems to grow back. Or if a girl is shaving her legs, there is more hair covering her legs. Yes, this happens. But that is simply because they haven't completely developed their body's full hair-growth. It will not be until full adulthood that all hair that is going to appear will appear. The process of maturing into an adult does not happen in one day. It takes years, and new hair growth will continue to occur over the course of those years. So when that boy or girl begins shaving at an early age (in their pubescence), they will be shaving off more hair by the time they have reached adulthood. But after that, they should notice that they are shaving the same amount of area until they die.
Before I get to the second definition, let me make one quick note here. At any one time, only about 1/3rd of your hair is actively growing (getting longer). If you could keep track of one hair follicle at a time, you would see that it will grow for about a month, then stop for a while. While one is stopped, another will be growing longer. With thousands of hair follicles, it is extrememely hard to see this. I suppose if you wanted to draw a small square on your body and count all the hairs inside that square, then shave it, wait a couple of days, and counted how many stubbles show up, you might be able to appreciate this better. You would probably count about 1/3rd to 1/2 the original hairs. Give it a month or so, though, and your count would start getting real close to the original.
Okay, for the second definition of thicker that people seem to think about when shaving off hair. People usually tell me that new hair is not as soft as it was before they shaved it, it tickles or scratches. So, they think, the hair must be thicker (in diameter). Wrong again (sort of). It is because the stubble is so short that it doesn't feel as soft. A hair needs to get to a certain length before it bends in all different directions, and therefore feels soft. If you take a long twig and a short twig from a tree, and try bending both, you will notice that the longer one seems to bend more. If you really want to get involved with the expiriment, you will notice that if you look at a small length of the longer twig (say the same length as your shorter twig) it only bends about the same amount as the shorter twig. It is because it is longer than the other and there's more of it to bend. This is what is happening with hair. As stubble, there is a limit to how much it can bend and twirl, and whatever else you want to do with it. It's still pretty much prickly.
Okay, I made a statement above that you may have noticed. It was the "(sort of)" after I said "Wrong again" to the diameter of hair. This refers to the diameter of hair at its root as opposed to the diameter at the end. Microscopically, especially on longer hair, the strand gets thinner and thinner the further out to the end you look. This is normal. Hair is made up of compacted dead cells that tend to flake off over time, especially when you wash or play with it. I have heard that the thickness of a strand of hair can be as much as 1/2 the thickness at its end as it is at its root.
Okay, I think I've said enough about hair. Now I need to go take a shower and shave before I go to work.