How does the hot gel hand warmer work?

I am not exactly sure what you mean by "the coin," but I think you are asking about the specific type of hand warmer(instant heat pack)  that contains a metallic trigger inside the packet. Such a packet contains a supersaturated solution of a salt. That is to say, the solution contains more of the salt than could have been dissolved into it at room temperature.

 

This means that the system is prone to spontaneous crystallization. Disturbing the metallic trigger provides nucleation points and the crystallization occurs. Heat is generated because the crystallization of the given salt is exothermic. (Some salts have an endothermic crystallization, which would instead produce a cooling sensation.)

 

The chemical warmer(instant heat pack) can be reused if it is boiled and then cooled. Boiling provides the heat that is needed for the solute to again become fully dissolved, and the gradual cooling results in the supersaturated state that is required to later produce heat.