What causes the energy produced by the motion of molecules to manifest as heat? Why is heat the resulting form of energy when molecules are in motion? Is this just an intrinsic property or is there a deeper explanation as to what causes this?
 A: Heat is actually not the motion of molecules. That is called internal energy, specifically, internal kinetic energy. Internal potential energy is that associated with intermolecular forces.
Heat is actually one of two forms of energy transfer. It is due solely to a temperature difference between substances. The other form is work (mechanical energy transfer). 
Heat transfers the kinetic energy of the molecules from a substance at higher temperature to the molecules of a substance at lower temperature. A good pictorial of the concept can be found here: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper.html
Hope this helps.
A: Maybe your question stems from a common confusion about the meanings of "heat" and "temperature".
The notion of temperature comes from our skin's perception of warmth.  Careful science showed that our perception of high or low temperature correlates to the amount of random molecular motion in an object, whether it is solid, liquid, or gas.  Thermometers were invented to provide a consistent measurement of temperature, independent of our perception.
Further careful science showed that the amount of energy required to change the temperature of an object by a given amount depends on the composition of the object as well as the mass of the object.  The portion of internal energy comprising random molecular motion was termed "heat".  So, the term "heat" is just our name for the random-motion component of energy.   So, asking why heat is the result of (random) molecular motion, then, is a bit like asking why your username is JuliaW.  
On the other hand, you may really be asking why we perceive increased random molecular motion as corresponding to a higher temperature.  But that question, too, is nearly unanswerable: it's akin to asking why we perceive a certain range of wavelengths as "red".  The fact that we have temperature sensors in our skin, though, does have an explanation but it's in the realm of evolutionary biology rather than hard physics: it's because it gave our ancestors a survival advantage.
A: Heat is the flow of energy, not the energy of the molecules.
If there is a bunch of them in a region, with more energy than the average, energy tends to spread. Or because the molecules spread themselves or because they collide to others, exchanging energy. 
