I was first seriously exposed to the concept of grounding when I was apprenticing in Shamanism at the Church of Earth Healing in Athens, OH.  If you are going to be blasting off into other dimensions on one sort of shamanic journey or another, a solid grounding in this physical reality is necessary.  Over the 20 years I have been studying and practicing healing, each school or teacher described grounding in a slightly different way.  Most described a common sort of presence in the here-and-now and a subtle energetic and spiritual connection to the earth.  After 4 years as a student and 10 years teaching at BBSH, I had plenty of not-so-subtle experiences of grounding.  Grounding came to mean a tangible surge of pleasant and powerful energy that would fill my body, especially my lower belly (tan tien).
However, despite years of doing grounding exercises, I was not prepared for the simplicity and elegance of the teachings about earthing that are now emerging.  Firstly, researchers have begun to accurately describe grounding from a scientific perspective.  This can be mostly summed up in one sentence: earthing supplies the body with electrons.  As Dr. Thomas Levy has said, all antioxidants supply the body with electrons, and all toxins steal electrons from the body.  If you haven’t seen any of the videos by Dr. Levy, this might be a good time to view some of those.  So, in essence, the earth offers a limitless supply of electrons, which operate as antioxidants and help neutralize just about any poison.  As we know, antioxidants are one of the primary treatments for damage due to radiation exposure.
Earthing can be accomplished simply by standing on the earth barefoot, swimming in a natural body or water, or, indoors, by using specifically designed grounding mats that are connected to the earth by a grounding wire.
Here is a summary of some recent research into the health benefits of earthing:
David Wolfe gives a live demonstration of grounding: