Here's a simple cross-section of the stomach (from here):
The stomach accomplishes much of its function by mechanically breaking down the swallowed food particles and mixing them with acid and enzymes into a sort of slurry. To do this, there are three major layers of muscle surround the stomach - from the outside, the longitudinal layer, the circular layer, and the oblique layer. The stomach also has two holes in it - the gastroesophageal opening, coming from the esophagus with the swallowed food/saliva mix, and the pylorus, where the food/acid/enzyme slurry exits into the duodenum, which is the beginning of the small intestine.
Due to the three layers of (rather strong) muscle, the stomach doesn't have a lot of expansion capability once it is filled completely to capacity. Fortunately, this almost never occurs (despite how we may feel after a large meal) because material is always leaving the stomach on its way to enzymatic digestion in the intestines. Additionally, once the stomach is filled to a certain extent, hormones such as leptin are secreted that give you the feeling of being sated, or full, triggering the brain to make you stop eating.
Of course, as we can see with the current epidemic of obesity around the world, the stomach can change its size over time. However, this is a rather slow process (weeks to months to years) of adapting to continuously consuming large meals.
But what would happen if you completely ignored these internal warnings, or were being force-fed, or whatever? Instead of rupturing (the biological equivalent of "exploding"), food would most likely be expelled either into the small intestine or back into the esophagus and back up the way it came down, i.e. causing vomiting.
For this answer to be complete, however, I should point out that stomach or gastric rupture is technically possible, although it is an extremely rare occurrence. Cases have been described in the literature (1, 2, 3, etc.), but they are rarely correlated with overeating or eating disorders like bulemia. In cases that are, other pre-existing conditions such as gastric ulcer have been found on autopsy that likely weakened the stomach wall in the first place. The bottom line - if you're an otherwise healthy person, a single instance of massive binge eating is very unlikely to end in stomach rupture, as vomiting will most likely begin well before you are in significant physical danger.