Actually, it depends how you mean by "grow back thicker".
When hair starts to grow naturally of its own accord, it starts out very fine and light, and it gradually tapers out and darkens to full width and colour at the root as each hair grows longer.
When you shave it off, it continues to grow, but this time the tip growing out of the follicle is already full-width and color -- since you clipped it off near the fat end, at the root -- and it hasn't been exposed to bleaching effects of the sun yet, so each entire hair continues to grow out with maximum thickness and darkness from root to tip.
So yes, shaving makes hair appear thicker, coarser and darker, but it doesn't actually make the maximum thickness of each hair any fatter than it's ever been; it just allows each hair to grow back at your pre-existing maximum thickness for the entire length, rather than tapering and lightening towards the tips.