I'm trying to get an intuitive understanding of what entropy is and, frankly, I'm starting to get desperate for the click in my mind that allows me to connect all that stuff. This is my current understanding: internal energy accounts for the total amount of energy contained in a system; alternatively, the amount of energy that would be needed to reproduce that system in that particular place and state. Thus, the equation dU = dQ + dW makes complete sense. Or even dU = KE + PE; PE including all different kinds of potential energy. However, I can't make sense of the equation dU = TdS + dW.

Why is the change of entropy related to the change of heat? Does an increase of entropy produce an increase of heat? It does. But why? Does it mean that a system which undergoes a decrease of entropy (at the expense of its surroundings) requires less energy to reproduce? Put it in another way: does the second law mean that, as the entropy increases in an isolated system, every system is increasingly rising its internal energy?

I'm thoroughly confused and I'd really appreciate some help from someone with a clear picture about this. Thanks a lot and have a nice day!