Examples of the moduli space of X giving facts about a certain X

The easiest example I can think of is the natural incidence correspondence between $\mathbb{P}^3$ and the parameter space of cubic surfaces. This can be used to show that every cubic surface contains a line; from this it follows easily that every smooth cubic surface contains exactly $27$ lines.

Another example is the moduli space of stable maps constructed by Kontsevich; this parametrizes certain maps from curves to (to stick with a simple case) $\mathbb{P}^2$. It can be used to answer the following question: given $3d-1$ points in $\mathbb{P}^2$ in general position, compute the number $N_d$ of rational curves of degree $d$ passing through these points. It turns out that the values $N_d$ satisfy a certain recursive relation which allows you to compute all these numbers starting from the obvious $N_1 = 1$ (through $2$ points passes exactly one line). You can find the formula here; it yields for instance $N_2 = 1$ and $N_3 = 12$.

Yet another example, again more elementary is the following. The Grassmannian $G = \mathop{Gr}(1, \mathbb{P}^3)$ parametrizes lines in $\mathbb{P}^3$. The computation of the cohomology of $G$ allows you to compute the number of lines which are incident to $4$ fixed lines in general position (it turns out this number is $2$).

Here is one example.

Let $M^3$ be a hyperbolic manifold. Consider the moduli space curvature $-1$ metrics on $M^3$ modulo $Diff_0(M^3)$. This is a point. Conclusion: every diffeomorphism of $M^3\to M^3$ is homotopic to an isometry.

You have chosen an example where the moduli space (a projective space) is a homogeneous space. So, geometrically, all the objects it parametrises are "the same", and the nature of the moduli space merely confirms that.

Perhaps it should be said first, therefore, that moduli spaces are not always homogeneous spaces. Not all points on the moduli space look the same, and therefore questions arise. This is seen classically for elliptic curves, where the typical automorphism group (preserving the identity) of an elliptic curve is of order 2, but in a few cases it may be of order 4 or order 6. Does this show up in the moduli space? Yes, when you construct it in the classical way from a fundamental domain in the upper half-plane. Moral: if there are "special" points in the moduli space, there is a geometrical reason they are special.

There are actually three levels to look at: the structure of the moduli space qua space (manifold-like, let's say, for complex geometry); for sophisticates using scheme theory the so-called infinitesimal structure in the sheaf given on the space; and the "moduli" themselves, such as the classical j-invariant, namely the parameters used to describe the space. It depends from what direction you are coming, but certainly for arithmetic special values of the moduli read back in an interesting way.