How do charge carriers move thermal energy? (Peltier effect) I am having hard time understanding how the charge carriers (electrons and holes) are able to move thermal energy. 
I am on a high school physics level, so I will probably have a hard time understanding it if there are advanced quantum physics involved. 
 A: The heat of a solid is related to the energy contained in the solid. Hotter solids have more energy in them.
Where is the energy in the solid? It's in the atoms and charge carriers moving/bouncing around. If something is hotter, the atoms and charge carriers are moving/bouncing around more and faster.
Say you have a bar, and one end is hotter than the other. The electrons at the hot end are moving faster (on average) than the electrons at the cold end. Because of this, you'll have more electrons moving from the hot end to the cold end than from the cold end to the hot end. These faster moving electrons take their energy with them, thus heating the cold end. (It's more complicated than that, because electrons won't get from one end to the other without scattering a gazillion times, but they can still transfer energy this way.)
Heat can also be transmitted by the atoms jiggling around. They can't move far because other atoms are in the way, but they can still transfer energy to their neighboring atoms by moving in sync with them. These waves are called "phonons". In metals, charge carriers transfer a lot more heat than phonons, but in semiconductors and insulators, phonons transfer the bulk of the heat because there are many fewer charge carriers.
A: Let's talk about semiconductors which are a bit easier to understand for this problem. Let's say we have a metal plate that we want to cool down. We attach an n-type semiconductor and a p-type semiconductor to different places on the  metal plate, and then force a current through that goes from the n-type to p-type semiconductor, through the metal.
Due to the Peltier effect, a cooling action happens in two places: once where the n-type semiconductor connects to the metal (where current is going into the metal, i.e., electrons are being forced out of the metal), and once where the p-type semiconductor connects to the metal (where current is going out of the metal, i.e., holes are being forced out of the metal).
Overall, we are forcing electrons out of the p-type semiconductor, passing them through the metal, and then forcing them into the n-type semiconductor. This requires energy, since the electrons we pull out of the p-type semiconductor come from below the Fermi level of energy (they come from its valence band), and the electrons we add to the n-type semiconductor have to be pushed up above the Fermi level (to be put into its conduction band).
This energy needs to come from somewhere, and where it comes from is the metal. Hence, the metal loses energy and cools down.
Basically we are evaporating electrons out of the metal into the n-type semiconductor, and evaporating holes out of the metal into the p-type semiconductor. Both of these cause evaporative cooling.
A: Well, it is kind of complicated, and not high school material actually :)
But, electrons/holes do transfer heat. They move around in a solid, bounces into stuff, and from that give of heat - if we have to keep it really simple.
Mostly though, the basic heat contribution (At least in your scenario with the Peltier effect an such) comes from lattice vibrations, or phonons. At specific temperatures, the heat comes from these phonons (Which I suggest you read about on Wikipedia, since it's a bit wierd), and the heat contribution from electrons/holes are pretty much negligible. But, at higher temperatures, you actually destroy the phonon contribution, and the electrons/holes take over, and the main carrier of heat is them.
So basically: Heat IS transferred by the carriers, but phonons are actually the main reason - unless you have high temperatures.
But as I said, phonons is a hard concept, and involves quantum mechanics to fully understand.
