As the temperature falls, the share in the upper level dies away as e^{-\varepsilon/kT}. Once kT drops below the spacing of the levels, there is nothing left to lift anything up with.
The figure is that change. The lower the temperature, the more the particles gather in the lower level. As T approaches 0, every one of them is in the ground state.
The state becomes a single state. Entropy was a count of states, so once we are here there is no avoiding saying what becomes of it.
If every particle is in the ground state, there is one arrangement available. W = 1, so S = k\log 1 = 0.
The figure is S(T). Lower the temperature and it falls to 0. This is the third law of thermodynamics, found by Nernst from experiment, and returned by statistical mechanics straight out of counting.
What matters is that this 0 is not a chosen reference point. It came out of a count, so it is the same 0 for any substance and any pressure. Entropy, unlike energy, therefore has an absolute value.
The ground state need not be single. If there are g states sharing the same lowest energy, then for N particles W = g^N.
The entropy is then S = Nk\log g, which is not 0. Of the two curves in the figure, the one that does not reach the bottom is that. It is called residual entropy.
Ice is the example. The oxygen positions are settled, but freedom is left in how the hydrogens are placed, so it never falls all the way to 0. The value Pauling counted agreed with the value measured. What the third law says is not that entropy becomes 0, but that it becomes a value fixed by the count of ground states alone.
If S approaches 0, then the heat capacity, which comes from its slope, must approach 0 too. Were a heat capacity to remain, S could not finish falling.
The two curves in the figure are that. Lattice vibrations fall as T^3, the electrons in a metal as T. These were found separately in the pieces on the Debye model and the Fermi gas. The derivations have nothing in common, and both go to 0.
Classically the heat capacity was a constant, independent of temperature: equipartition returns 3Nk. That it fails at low temperature is because the levels come in steps. The third law does not hold without quantum mechanics.
There is a way of cooling called adiabatic demagnetisation. Apply a field to line the spins up, then cut off the exchange of heat and remove the field. Removing it lets the spins scatter again, and the temperature falls in exchange.
The figure is that, repeated. Two curves of S(T) are drawn, one with a field and one without, and the path descends between them in steps. Moving across is applying the field; dropping down is removing it.
But the two curves meet at the same point at T = 0, because both fall to 0. So each step down is smaller than the last, and no finite number of repetitions arrives at 0. That absolute zero cannot be reached is exactly this shape. Even being unreachable follows from the one fact that S falls to 0.