In the previous piece the transition was where the number of answers went from one to three. That line of argument looks particular to the Ising model. But all it used was that nothing in the expression tells up from down, and one number m.
What changes at a phase transition is usually a single quantity. For a magnet the magnetisation, for a liquid and its vapour the difference of density, for a crystal the displacement of the lattice. Each is 0 on the hot side and not 0 on the cold side. Such a quantity is called an order parameter.
The left of the figure is the symmetric state, the right the broken one. What Landau did was to stop solving models and instead write the free energy down as a function of the order parameter alone.
Near the transition m is small, so F may be expanded in powers of m. What can be written is F = F_0 + am^2 + bm^4.
There are no odd powers because turning every spin over must leave F unchanged. If m and -m have to give the same value, m and m^3 cannot survive. It is the symmetry that settles what may be written at all.
And b has to be positive. Make it negative and F falls without end where m is large, leaving the system nowhere to settle. The only coefficient free to move is a. The curve in the figure is what happens as that a is moved.
While a is positive, F is smallest at the single place m = 0. Once a turns negative, m = 0 is no longer the smallest, and the minimum splits into two, left and right.
The figure is F(m) as a is moved up and down. Setting a = T - T_c puts the splitting exactly at T_c. That one assumption, that a changes sign with temperature, is all that has been put in.
Not one model has been solved. We wrote what could be written and let a coefficient change sign. And still the same shape came out as from solving the \tanh in the previous piece. The look of a transition does not lean on the fine detail of the model.
Differentiating F by m and setting it to 0 gives 2am + 4bm^3 = 0, that is m^2 = -a/2b. Putting in a = T - T_c gives m \propto (T_c - T)^{1/2}.
That 1/2 is called a critical exponent and written \beta. The curve in the figure is it, and it lies exactly over the shape the Ising model gave.
Nowhere have we used what a or b actually are. All we used was the symmetry and the fact that a changes sign. If that is so, \beta = 1/2 ought to be a value that no model can change.
Experiment, however, returns about \beta = 0.33. The two curves in the figure are that difference. The smaller the exponent, the steeper the rise, so just under T_c the measured one passes above the mean-field one.
The reason it fails is that m was made one number with no dependence on place. Near the transition, regions that agree and regions that do not fluctuate enormously, and the fluctuation is not a small disturbance around the average but as large as the average itself.
Whether fluctuation matters is settled by the dimension. The higher the dimension the more neighbours there are, and an average will do. Above four dimensions mean field is right; in three it is not. How to count what it misses is what renormalisation goes on to do.