Reason for thermal motion I've been brought up to believe that the higher the temperature of a material, the faster the "random motion" of the molecules in that material will be.  Consequently I think of density (eg. Of uncontained gas) dropping with increased temperature because the molecules are banging against each other at higher speeds, forcing the molecules further apart etc.
Can someone explain a mechanical/physical reason as to why molecules should bounce around, and why temperature would make this physical motion more aggressive/forceful?  
Take the example of an exothermic chemical reaction - two molecules react, generating heat. How did the chemical reaction increase the velocity of the molecules (without invoking the circular argument "the temperature increased")?
 A: An increase in temperature does not necessarily reduce density. For that to happen the volume needs to increase. But an increase in volume, and decrease in density, can take place without any change in temperature. Rather, for a fixed mass and volume of an ideal gas, increased temperature is noticed as an increase in pressure :
$P=nkT$
where $n$ is number density of the gas molecules. 
Your 1st question (why does an increase in temperature make atoms move more violently?) has a trivial answer. What we measure as temperature is the effect which is caused by the violent random motion of atoms. For example, we measure an increase in temperature by the increase in volume of some mercury or alcohol in a sealed tube, or of some gas in the above piston-chamber. 
Your 2nd questions is more interesting and more difficult to answer : What starts this motion in the first place? How does chemical or nuclear energy turn into kinetic energy? Saying that mass is energy and pointing to the equation $E=mc^2$ does not explain the process(es) which are involved.
One possible answer is that the rearrangement of electrons in a reacting molecule releases photons with energies of a few $eV$ which is 2 orders of magnitude larger than typical kinetic energies of molecules, which are $\frac32kT = \frac{1}{40}eV$ at room temperature. This explanation is appealing because some reactions, such as the combustion of magnesium, release intense amounts of light. However, the momentum of a photon is orders of magnitude smaller than that of an atom. So this cannot be the explanation even for magnesium.
The answer must be that excess chemical energy is, in almost all cases, released as the kinetic energy of the products and/or catalysts. The difficulty with an exothermic reaction of the type $A+B \to AB$ is that an increase in kinetic energy defies the conservation of momentum. Interaction with a 3rd molecule is essential. The intermediate state is highly unstable but exists for a short time. If the gas is under high pressure then a collision with another molecule becomes likely within the lifetime of the excited state. Reactions of the type $A+B \to C+D$ do not have the same problem.
In nuclear fission, the electrostatic repulsion between daughter nuclei is held in check by the strong nuclear force. Once the nuclear force is overcome, it is easy to see how the electrostatic potential energy is converted into the kinetic energy of the fission products.
See also :
How does the breaking of chemical bonds turn into kinetic energy?
The exact mechanism of energy release durning bond formation on the atomic level
A: In the case of an ideal monatomic gas, random-motion kinetic energy per particle is the way we identify a temperature.   A diatomic gas, or a chemically active gas will have a more complex thermal behavior (because there are other energy-containing degrees of freedom).   When that hypothetical second gas comes into equilibrium with the simple monatomic gas, it is (by definition) at the same temperature.   
There is no reason why molecules should bounce around freely.  A solid, or liquid, can also come into thermal equilibrium with the hypothetical ideal gas, and when it does, it is being hammered by all the gas molecules that come into contact with it.  So, it is moving (vibrating, quivering, ringing like a billion bells).     That, in a nutshell, is why temperature is expected to make internal motions more energetic.
If the solid weren't jostling around, if it were just taking damage from the gas, it would COOL THE GAS.  However, it is in equilibrium, so it has to be moving, bouncing gas molecules into higher velocity just as often as a gas molecule loses velocity by rebounding with a dull 'thud'.
A: "Can someone explain a mechanical/physical reason as to why molecules should bounce around, and why temperature would make this physical motion more aggressive/forceful?"
Temperature is a measure of the average kinetic energy of a group of atoms or molecules. The higher the temperature of, say, a container of air, the greater the average kinetic energy of the air molecules in the container.
What could increase the average kinetic energy of the air molecules in this container? Radiation could be one thing. If you could shine a "gamma ray flashlight" on those air molecules, the gamma rays would tend to increase the kinetic energy of the molecules they collide/interact with. Energy is being transferred from high to low, the way a pool shark breaks the starting triangle.
Again, temperature is just the "electronic readout" of how energetic the molecules/atoms are on average.
