Shaving doesn't cause hair to grow back thicker.

Your hair's thickness depends on your genetics and your hormones. Whether you shave makes no difference.

It also doesn't make it grow back faster or make your hair rougher. What really happens is that as your hair grows it tapers off at the end. The ends of your hairs are thinner and appear less coarse. When you cut that part out, what's left is the thick part of your hair. So it seems like it got thicker and coarser.
(Sources: 1, 2)