I was completely unaware of the concept of hydrogen-rich water, and although there is a lot of misinformation and quackery around it, a quick search on Google afforded an article (whose quality I cannot vouch for). It would seem that hydrogen-rich water is nothing but regular water with dissolved hydrogen gas. You could simply pump $\ce{H_{2(g)}}$ into some water, but there is a smarter way to do it, by producing the hydrogen in situ via the addition of reactive metals such as magnesium, according to the following reaction:
$$\ce{Mg_{(s)} + 2H2O_{(l)} → Mg(OH)2_{(aq)} + H2_{(g)}}$$
Producing the hydrogen directly inside the water also seems to temporarily supersaturate the solution and increase the amount of hydrogen dissolved, because the equilibrium solubility of hydrogen gas in water is tiny, hydrogen gas is only soluble at around 1.6 mg per kg of water at 20°C. The exact mechanism by which hydrogen-rich water has medicinal properties I do not know, and that would probably be best answered on Biology.SE if you're curious. It is conceivable that the dissolved hydrogen gas could act as an antioxidant, though relying on the distribution of a gas across the body seems a bit strange (there are gaseous substances known to be important for our metabolism, such as $\ce{NO}$). I wouldn't rule out the possibility of it being nonsense before reading into it a lot more, though.