Short answer: The charges themselves don't dissipate. The potential energy we stored in the capacitor from the difference in charge is converted to other types of energy, some of which is heat, as the electrons (negative charges) move through the circuit.
If you "short" the two leads of the capacitor, you touch the metal leads together. Those metal leads have a very small resistance. It's so small that we say that is effectively a "short," even though it technically is not. When you put a voltage across a resistance to push a current through it, some of that potential energy (the voltage) eventually gets converted to heat energy.
Electric space heaters generate heat this way; they pump current through what is essentially a big old resistor for the purpose of turning it into heat. Light bulbs are similar, but they dissipate the energy as both heat and light. Though in our case, it is not purposeful, because there is a small amount of resistance in just those metal wires, it acts like a tiny version of the space heater, and some of our energy is dissipated as heat.
Note: When I say "dissipated as heat," really the energy is just being changed from one form to another. We say that it's being "dissipated" because it's being converted to a form we don't want it to be, and so it's "wasted" (e.g. if we had a light that didn't generate any heat, much more energy would be being converted to light, so our batteries would last much longer).
Long answer:
A capacitor is effectively two parallel metal plates. It stores a surplus of negative charge (electrons) on one side, and we say that it stores a positive charge (a relative lack of electrons) on the other plate. This difference in charge results in a voltage, which is a difference in potential energy between the two plates. When there's a difference in potential energy, it can be turned into mechanical energy. In our case, the potential energy is this voltage across the capacitor, and the mechanical energy will be the physical movement of the electrons through the circuit (i.e. an electrical current).
(Other examples: Elastic potential energy: use a mousetrap spring to power a mousetrap car; Chemical potential energy: burn gas to break the bonds between the atoms to release the energy and power a real car; Nuclear potential energy: break apart the atom itself to release the energy stored within it to power a real city)
The way we build up this surplus of electrons is by using an outside voltage source like a power supply or a battery to push a whole bunch of electrons to one side of the capacitor.
___
|
| <---wire connected to positive voltage source (electrons got pushed out this way)
|
|
+++++++++++++++++++++
+++++++++++++++++++++ <---positive charge (lacking electrons) Note: not entirely lacking electrons. Just some got pushed out.
___________________________
<---metal plates (not touching)
___________________________
-------------------------------------- <---negative charge (surplus electrons)
--------------------------------------
|
|
|<---wire connected to ground (electrons got pulled in this way)
|
____
__
_
Once we have this buildup of different charges, resulting in a difference in the potential energy, we can put it into a circuit with a resistor and power the circuit from there.
_____________
|+ |
CAPACITOR RESISTOR
|-___________|
<--current--
Once we put the capacitor in the circuit with the resistor, the natural reaction is for the negatively charged particles to push away from each other and move toward the positively charged side. Since the two metal plates of the capacitor are separated, the electrons will move from the negative side to the positive side, through the resistor, until there is no difference in the build-up of charges on either side of the capacitor.
These moving electrons have kinetic energy associated with them, as they are particles that have both charge and mass (like a moving boat has energy associated with it). A resistor, as the name implies, resists the movement of these electrons as they are getting pushed through the resistor. If that boat from before was an ice-breaker in the arctic, plowing through some ice, it clearly has enough energy to keep moving through that ice, but the ice is slowing it down a bit, and it's imparting some of that energy to the ice by hitting it, breaking the ice.
This is an example of an object being resisted, imparting kinetic energy to what's resisting its movement. Electrons moving through resistors is an electrical version of that. The electron is running into a whole lot of tiny electrical fields from the particles in the atoms in the resistor, and it's getting slowed down and pushed around. Just like the ice-breaker pushes back against the resisting ice, the electrons push back against the resisting fields ("Something something Forces, something something Equal and Opposite Reaction" - Isaac "Big Daddy" Newton).
Since energy can't be created or destroyed, the energy has to be transferred completely into some other form (or forms). We're releasing the potential energy (the voltage) into kinetic energy (the motion of the electrons), and some of the energy from the moving electrons gets transferred to the material of the resistor as some other form of kinetic energy. This is most often heat, but can be light (e.g. light-bulbs or something that's "red hot") or motion (e.g. motors) or sound (e.g. speakers), for example.
As we said before, even when you "short" the leads of the capacitor together, there is a tiny bit of resistance in the metal leads, so "shorting" the capacitor still gives you the situation described above, where the energy from the voltage is eventually converted into heat energy.
Side Note: As more electrons move from one side of the capacitor to the other, the difference in charge drops, so the difference in energy levels across the capacitor drops, so the voltage drops. That's why we see the voltage going down in that voltage vs. time graph in my previous post.
<EDITED to show where surplus electrons go/come from while charging a capacitor in Fig. 1>
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