The first time I barbecued a brisket, I sliced into it immediately after taking it out of the WSM. I was surprised to see a gush of juice run out all over the cutting board, and afterward I wondered why the meat was so dry. My experience is an example of why it's important to let large cuts of meat rest for 10 to 30 minutes after cooking.
In the book How to Cook Meat, authors Chris Schlesinger and John Willoughby state that as meat proteins are heated during cooking, they coagulate and squeeze out some of the moisture inside their coiled structures and in the spaces between the individual molecules. The cooking heat drives this liquid toward the center of the meat. In CookWise, author Shirley O. Corriher reveals a bit more detail: as meat proteins cook, they begin to shrink. Up to 120*F, the proteins shrink in diameter only and there is little moisture loss, but above 120*F the proteins also begin to shrink in length, which really puts the squeeze on moisture. By 170*F, most of the moisture can be squeezed out of a lean piece of meat.
As meat rests, this process is partially reversed. The moisture that is driven toward the center of the meat is redistributed as the protein molecules relax and are able to reabsorb some moisture. As a result, less juice runs out of the meat when you cut into it. Willoughby claims that if you cut into meat right away, almost twice as much liquid is lost than if you let it rest before carving.
How long should meat rest after cooking? It depends on the meat. Most well-documented recipes will specify the resting time. The length of time usually depends on the size of the meat. A steak might rest 4-5 minutes, a whole roasted chicken 15 minutes, and a standing rib roast 30 minutes. During the resting period, cover meat loosely with aluminum foil to prevent the surface from cooling off too fast.
Thin cuts of meat, like pork ribs, don't need any rest at all. They're best eaten immediately.
On a related note, residual heat continues to cook thick or large cuts of meat during the resting period, especially those that have been grilled or roasted at high temperatures. It is not uncommon for internal temps to increase 5-15*F during this time. You must take this additional cooking into account when deciding at what temperature to take meat off the cooker. For example, when roasting a prime rib at 325*F and targeting a final internal temperature of 135*F, remove the meat from the fire at 120-125*F, cover loosely with foil, and let rest for about 30 minutes. The roast will rise to your final internal temp during the rest.
Meats cooked at lower temperatures will experience a rise of only 1-2 degrees internal temperature during the resting period. A butterflied chicken cooked at 225*F is an example of meat that won't rise much during a 5-10 minute rest before carving.