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Basically, it's all about the gas, or at least in this case stuffing that gas into a bottle (watch carefully, and you'll see the watermellonistas put the dry ice or liquid nitrogen into a bottle, and then the bottle goes into the mellon). The volume occupied by a given amount of a liquid or solid increases several hundred times when it becomes a gas. If you do that in a sealed bottle, it becomes pressure, and when that pressure reaches the maximum limit of the bottle it bursts...violently. Pretty simple, eh. Oh, and yeah, those exploding bottles do explode with enough force to cause injury. So be careful if you try this at home, particularly since it's almost impossible to know when it's going to go off.
In this case a typical PET bottle will burst at about 12 atm (about 1250 hPa). That pressure can be achieved with about 25 cc of liquid nitrogen or 30 grams of dry ice in a 2 L bottle. I guess plastic bottles do not generate shrapnel when they burst, but I sure wouldn't want to be holding that bottle when one does.
As for the Mentos in Coke...It has nothing to do with either mentos or coke. The CO2 that's dissolved in the water of any carbonated drink (beer included) can be driven out by adding something else to that. Most likely the candy shell provides nice sites for CO2 nucleation (bubble formation), and that increased nucleation causes the CO2 to rapidly degas from the water. More-or-less, it's like shaking a bottle of soda, with your thumb over the mouth.
