It's a unique experience in the U.S. but common in Asia. Many options on broth in which to cook your veggies and meats. My favorite was the tomato-based broth, which added nice flavor to the otherwise bland veggies. For the unitiated, I would advise that you first try Chubby Cattle with someone who knows the ropes and can guide you on how to make your own dipping sauce and what meats to order. It can be a bit overwhelming if you come in cold turkey, though the staff is friendly and can help a lot.
Biggest danger is choosing too many items with no knowledge of the price. It gets expensive fast and a glance at your bill can end up inducing acute indigestion. It's one if the reasons I tend to avoid dim sum.