What does resting mean

There is, as you've seen, no universally precise definition. Broadly, though, "resting" refers to allowing heat to diffuse through the food. Although the burgers are "still cooking" once placed on the higher rack, the amount of heat being applied is nowhere near as high as when they were on the grill, and following the resting the temperature differential will be lower than when they were first taken off the grill.

This is a good scientific approach to looking at the concept of resting meat. Resting commonly means to remove the meat from the cooking surface and allowing it to sit, untouched, for some time. We do know that "carryover cooking" is real and that items continue to cook once removed from the heat. This needs to be factored into your preparation. I would encourage you to read the information I linked above. His bottom line:

Roughly speaking, comparable amounts of moisture evacuate during cooking as dribble out after slicing. If the final temperature is below 130F, collagen barely shrinks and there is no difference between resting and not resting. If the final temperature is above 145F, the rested meat will exude more juices than the non-rested, but... Resting meat merely shifts when and where the juices are exuded- the total loss is about the same if you allow the juices to be reabsorbed. Thicker meats benefit slightly from resting- mostly by making them easier to slice and resulting in a more uniformly cooked roast. But remember the thicker the meat, the more "carry-over", so adjust the cooking temperatures accordingly. Don't waste the juices! Incorporate them into a sauce or sop them up on the plate.

All foods contain a certain amount of water, even meat. Those waters, when heated up through cooking, begin circulating through the food, faster and faster, raising the interior temperature of the meat(or whatever). This is called convection. When cooked food is removed from the heat source, convection does not just come to a dead stop; it takes a while to slow down, and then, eventually stop. All the while the meat's interior temperature continues to rise and the food continues to cook (this is called "carryover" cooking). If you cut into meat while this convection action is still happening, all of the waters (juices) swirling around come flooding out. This water loss results in dry, overcooked meat. By waiting for convection to stop, those waters are reabsorbed into the food and it retains it's moisture. Also, if you wait until the internal temperature of your meat is where you want it to be before taking it off the heat, carryover cooking will overcook the meat.