In the microcanonical ensemble we chose the arrangement that made the entropy of an isolated system largest. But what we actually want to handle is a system at a settled temperature, and a settled temperature means exchanging heat with the surroundings, which is no longer isolated.
The small box in the figure is the system we want to watch, the large one the bath. Put together they are isolated, so what is made largest is their combined entropy.
We only want the small box, and yet the quantity we can say is largest belongs to the two together. Close that gap and the answer can be had from the system alone. That is what this piece does.
The combined entropy is S_1 + S_2. If the system holds energy E, that energy came out of the bath.
The bath is large, so losing E does not move its temperature. Its loss of entropy is therefore just \Delta S_2 = -\dfrac{E}{T}. Nothing about what is inside the bath needs to be known. That is the essential point.
Putting it in gives S_1 + S_2 = S - \dfrac{E}{T}. Multiply through by -T and it reads E - TS, which we call the free energy F. The combined entropy being largest and the free energy of the system being smallest are one statement said two ways.
Take as an example a system whose state is settled by one number x, where a larger x means a higher energy and, at the same time, more arrangements available.
The curve in the figure is F(x). The lowest place on it is the x the system settles at, at that temperature. Move x a little either way and F rises, so it does not leave.
What we are looking for is not the smallest energy but the smallest F. Choosing by energy alone is right only at zero temperature. While there is a temperature, the count of arrangements comes in alongside it.
The figure draws the two terms of F = E - TS separately. E rises to the right, -TS dips downward, and their sum is F.
Raise the temperature and the dip of -TS deepens, moving the lowest place to the right. When it is cold E wins and x stays small; when it is hot TS wins and x grows.
This shape has appeared again and again. The share in the upper level of a two-level system, order against disorder in the Ising model, whether a boundary enters a one-dimensional chain. Every one of them was E against TS, with the temperature moving where the line falls.
F appeared because we held the temperature fixed and let go of neither the volume nor the particle number. Hold something else fixed and a different quantity is what becomes smallest.
Hold the pressure and it is the Gibbs free energy G = F + pV; allow particles to come and go and hold the chemical potential, and it is the grand potential \Omega = F - \mu N. The figure lists them, with more quantities held fixed as it goes down.
They are all built the same way. The single fact that the entropy of the isolated whole is largest, rewritten in the language of the system we want to watch. There are as many rewritings as there are quantities to hold fixed.