A possible mechanism may be through pressurization. Since liquid water is more dense than solid ice, high pressures will lower the freezing point of water. So, lets say the drinks are pressurized such that the freezing point of the soda is -5oC. Storing the drinks at about 0oC would not cause the soda to freeze. However, once the pressure is released, the freezing point of the soda returns to the normal 0oC, causing some of the soda to freeze.
(an interesting side note. The formation of ice from water is an exothermic process. So, these "self-chilling" drinks would actually slightly increase the temperature of the soda by causing ice formation!)