Energy in the Ising model for liquid-gas transition

I am teaching myself statistical mechanics using the lecture notes of Professor Leonard Susskind. I am confused about a statement ‘reducing $h$ reduces the density’ at page $555$. I know the statement is right from intuition since $h$ changes $bar{sigma}$. However, I cannot see it from the formulas. And I get more and more confused when I reread this chapter. Which represents the fluid? The particle or the void position?

In the last chapter, it’s said we can use an empty lattice with particles in or out to represent the liquid-gas transition. At default, the spins are pointed down with energy $-1$. The energy of the system can be written as

$$
E=-jsum_{links}sigma_nsigma_m+hsigma_n.
$$

Then it stated that when we introduce a particle into one void position. We need an energy of

$$
8j+2h.
$$

I have no problem with these numbers. Nevertheless, after I reread this part several times, it seems if we introduce a particle, the energy of this system increases. How can this ever be possible to suck any particles? On the other hand, $h$ here when it’s increased can create a lower energy empty position.

This lower energy empty position could be related to lower chemical potential. But I don’t know how this two be connected because whatever lower energy it creates, if you introduce a particle it increases the energy. Moreover, you lower more and introducing particles increase more. With the energy increased, as it increases higher when you increase $h$, I think it might reduce the density other than reducing $h$.

And more confusing thing is that the empty lattice is the ground state. Thus if I decrease the temperature, till at $T=0$, the systems has no particle at all.