The meat (whatever meat it is) is marinated and cooked over an open flame at your table, thus the "fire meat" meaning. When the meat is nearly cooked, it is cut into bite size pieces using a pair of kitchen scissors. Although I've eaten at many bulgogi restaurants and am capable of cooking and cutting the meat myself, I'm rarely allowed to do it. Koreans at the tables around us are always allowed to do it themselves but I think the waitresses assume that foreigners either don't know how or can't do it right.
There is a process to eating at a bulgogi restaurant. Among the many side dishes, you will always get a basket of lettuce or other leafy greens. You take a piece of meat and wrap it in a leaf along with a bit of ssamjang (red bean paste) and a bit of any other side dish you want, and then eat it. It's like a miniature lettuce burrito of sorts! Rice is usually not served at a bulgogi restaurant unless you specifically ask for it. With all of the grilled meat and vegetables, it's a very healthy and not overly filling meal.
"One cannot think well, love well, or
sleep well, if one has not dined well."