In the last step we use the fact that the system is isolated, i.e. there is no change of the heat. Irreversible processes make sure that the entropy actually always increases. If the universe is an isolated thermodynamic system, then its entropy must increase all the time. Since that is not at all like the universe we know, it follows that the universe is not infinitely old. Statistical mechanics is closely related to thermodynamics.
Statistical mechanics looks at the microscopic construction of thermodynamic systems.
Since there are a large number of identical microscopic particles, statistical arguments are used. Each molecule has its own distinct position in phase space, and with so many molecules about one starts to look at how many molecules are in a certain region of phase space, rather than where each molecule is. We neglect here the fact that air molecules are not all the same, we also neglect the fact that they have angular momentum as well. In our minds we would cut up the phase space into cells of equal size and count for each cell how many molecules have their state fall into that cell. We are building up a histogram, only that our cells are not intervals on the real axis, but six-dimensional cubicles in phase space.