In my opinion, there is no such thing as water veins as described in this article, but, it is a fact that there is water circulation underground.
In the natural environment, water can either circulate in porous mediums (like within the small spaces between sand grains in sandy soils) or as underground streams. There is also the concept of the water table and aquifer recharge and depletion.
This schematic provides a broad picture on the importance of groundwater
More info here as well.
Some regions composed of rocks, especially carbonates (karst) can have more often channelized underground streams, as the ground is not necessarily porous so water is kind of forced to flow where there is an opening (see following figure). (More info on karstic landscape here)
So that was a small overview on how water can circulate underground.
Then people getting sick from water circulating underground is not because of the veins per se, it could be because of moisture (or many other factors). But then living near a lake or any streams, any wet areas or in a badly insulated house could make people sick as well.
This is 'old wives' tale' (no offense - English is not my primary language I am not sure how to say) and is not really a thing as presented in the http://swissharmony.com/en/water-veins-neutralized site. There has been no new research in a medical paper about geopathy since 1940 (wonder why?). You are doing well in being skeptical! To conclude I am sure many people living in drylands would be more than happy to be 'sick' over a vein.