Edit: Contribution of enthalpy and bond formation
Edit 2: Changed the description of entropy
After thinking for sometime, I realized that I neglected to mention three things:
The Driving force of precipitation reactions depends on the charge and size of the ions which corresponds directly to the magnitude and order of the hydration shells formed upon dissolution of the salt.
The enthalpy as a driving force of the precipitation of the salt. (Remember the Gibbs Free energy equation has two terms, entropy and enthalpy)
Additionally, the formation of ionic bonds can be lower in energy (more stable) than the electrostatic interactions present in solution. The release of energy during salt formation is transferred to the surroundings as heat. This would be a negative enthalpy (heat flowing out of the system) and if large enough, would overcome any decreases in entropy stemming from the formation of highly order crystalline structures and drive the precipitation.
In this case it is probably the enthalpy term that is driving the reaction, since you are putting a ton of heat into the system to dissolve the salt, once the solution cools, there is a lot of potential energy that is stored in the position of the atoms in the system. This is then released as heat upon the formation of the crystal structures.
Entropy changes in precipitation reactions:
In order to make a supersaturated solution, you have to add energy to form hydration shells around the ions (entropy in the system is decreasing). The thing to remember is that these hydration shells decrease the dispersal of the ions relative to that in the crystalline structure for some salts. When you heat and mix the solution with excess solute, you are providing sufficient energy to form hydration shells. When the solution cools (Slowly), the hydration shells remain intact, however, the system is unstable and applying enough energy to the solution will disrupt the hydration shells and crystallization of the salt will occur. The large amount of heat that is released is coming from the increase in the micro-states available for the ions (gain in entropy) in removing the hydration shells around the ions. The hydration shells are less dispersed than the crystalline structures of the salt and that is why entropy increases during some crystallization reactions, making this process exothermic.
Here is why the disc snapping is important:
Even though this process is very exothermic, it still requires activation energy. The mechanical force you are referring to (i.e. snapping the disc or whacking the heating pad) is doing just that by simply transferring energy to the molecules, through vibrations, which provides the activation energy for crystallization.
Just as walking can cause energy transfers from the mechanical forces of your legs to the molecules of the floor (i.e. through friction) and increase the temperature of the surface, some of the force applied on the disc transfers to the solution and if it is enough, will overcome the activation energy barrier of the process and crystallization will begin.