Since heat transfers take place only between systems with different temperatures, explain why the third law of thermodynamics is supported by our current understanding of matter

The third law of thermodynamics focuses on the entropy of the system. According to this law, it states that 

"The entropy of a system at absolute zero is exactly equal to zero"

There is really no definite definition for entropy but differential equations. Since that is complicated enough, you can just think of entropy as the disorder in the system. Molecules engage in rapid collision at high temperature because of the high kinetic energy. The disorder is high. But if at very low temperatures, the molecules are very ordered or motionless. Hence, entropy approaches to zero.