Intuitive explanation why rate of energy transfer depends on difference in energy between two materials? The temperature of an object will decrease faster if the difference in temperature between the object and it's surroundings is greater.
What is the intuitive explanation for this? 
 A: I think it’ll be helpful to think in terms of the kinetics of the constituent particles. When an object is at a higher temperature, the kinetic energy of its constituents is higher. They are more in motion when compared to the ones with a lower temperature. 
Now if there’s a high temperature object in contact with a lower temperature one, there will be transfer of kinetic energy at the interface. Energy will be transferred in both directions except that the net transfer will be from higher to lower. This is because there are more ways for energy to be transferred from the higher to the lower. But as the temperature difference decreases, the rate of transfer from high to low and low to high are closing in. Until they reach equilibrium where the transfer of energy from either sides are now equal. Thus there will be no more net heat transfer on the average. 
A: You know that temperature is related to the microscopic kinetic energy of the atoms and molecules that make up a material. For simplicity, let's assume the two materials consist of monatomic ideal gases with one having a higher temperature than the other. Then the temperature of the two gases is a measure of the average kinetic energy of the gas atoms, which in turn depends on the speed of the atoms.
If the two gases are brought into contact with one another, at the interface collisions between the higher speed atoms of the higher temperature gas with the lower speed atoms of the lower temperature gas will transfer kinetic energy to the lower speed atoms. Those atoms will, in turn, move into the bulk of the gas and collide with others increasing their speed. Eventually when thermal equilibrium is reached the two gases reach some common intermediate temperature.
How quickly the temperature rises in the lower temperature gas will depend on how quickly the atoms at the interface move into the bulk of the gas and collide with other atoms raising the overall kinetic energy. All other things being equal, that penetration will be quicker the higher the speed of the atoms after colliding with the more energetic atoms of the higher temperature gas at the interface. That speed will greater the higher the speeds of the high temperature gas atoms, which in turn increases with the temperature of the gas.
Bottom line:  The greater the temperature difference, the more quickly energy is transferred into the interior of the lower temperature material and the faster its temperature rise. For more discussion of temperature I suggest you look at the Hyperphysics website.
Hope this helps.
A: Temperature is (to simplify it a bit), the average energy of the molecules, and the more energy each molecule has, the more it can transfer when it contacts molecules in the other object. So each object gives heat to the other proportional to temperature. The net heat transfer is the difference between how much energy the hot object is giving the cold one, versus how much the cold one is giving the hot one, which is proportional to the difference in temperature.
Imagine two friends who each day give 10% of their total money to the other. Since 10% of the richer friend's money will be a larger amount in absolute terms than 10% of the poorer's friend, the net change in the poorer friend's money will be positive, and it will be proportional to the difference in their money. If R is the richer friend's money, and P is the poorer's, then the poorer friend will lose 0.1P but gain 0.1R, for a total change of 0.1R-0.1P. We can factor out the 0.1 and get 0.1(R-P). 
This applies to conduction. For radiation, heat transfer is proportional to the fourth power of temperature, so net transfer is given by the difference in the fourth power of their temperatures.
A: An intuitive way to think about heat is as a rink of drunken roller skaters wearing perfectly elastic sumo suits.  They are skating around in random directions, with the one caveat that they always skate in a straight line until they bump into something, at which point they bounce like billiard balls.  The skaters have a random distribution of speeds, but the average speed is the "temperature" of the material.
If you have group A with a leisurely "temperature" of 5 mph (barely above walking speed) which comes in contact with group B whose "temp" is 6 mph, it will, at first, be difficult to even tell the temperature difference with just a visual inspection of the groups.  After all, they are all going close to the same speed to begin with.  However, the sumo skaters with the higher speed from group B will eventually start bouncing into skaters from group A, transferring some of their higher energy.  But because the delta is small, it will take a long time for average velocities to change much.
Now, imagine group C with a temp of 20 mph.  These skaters are hopped up on meth and are skating as hard as they can.  When they collide with group A, they will quickly deliver a big jolt of energy to the lethargic skaters.  In much less time, the extra energy from group C will get distributed to the slower members of group A, bringing up their average speeds.  More importantly, each collision between a fast skater and a slow one will produce a much larger average change in both their speeds, and this molecular transfer of momentum is what defines "heat flow".
