Since everyone talks only about induced voltage don't forget that the PUT (person under test) is also a plate of a capacitor.
I may have been harsh saying that they didn't understand what they were measuring, but yes, that was basically it.
The fact they can measure a smaller potential difference when connecting the system - their body - to a potential reference such as "earth" here, is a basic phenomenon. Also, their small "experiment" and misinterpretations of it may remind some of the famous Walter Lewin's experiment on Kirchhoff's laws...
Now whether this may have any impact on health is another story. I don't think there is any proof of that. I will be humble though - we are indeed bathed in increasing electromagnetic "pollution". Would I sign if I had to engage my full responsibility claiming that it has absolutely NO long-term effect on health? I would not. I have humbly no clue. But I wouldn't promote any weird pseudo-protection, or publish any panic messages on the topic either.
But as some people said here, walking barefoot on mats connected to "earth" can be a direct hazard. One thing you absolutely need to do is ensure there is a large enough impedance to the "earth" connection - in the order of several megohms - otherwise you could very well get electrocuted. As to walking barefoot close to high-voltage power lines, it's usually ok but could be a serious hazard if there is a fault in the installation. Oh, and that can also not be advisable when walking close to fields with electric fences.