A big feature of algebraic and complex geometry is that spaces come in continuous families, and those families can again be given the structure of algebraic or complex spaces. To fix ideas, let's say we have a compact complex manifold $X$. Following Kodaira, Spencer, Kuranishi and others we can say there exist complex manifolds $\mathcal X$ and $B$ and a proper holomorphic map $\pi : \mathcal X \to B$ such that $X = \pi^{-1}(b_0)$ for some point $b_0 \in B$. The other manifolds $X_b = \pi^{-1}(b)$ are then deformations of $X$.
If we suppose $X$ is Kähler then so are the close deformations of it as well. If we fix some extra structure or assume nice things, then Weil, Peterson, Siu and others have shown how to construct Kähler metrics on the base $B$, so it becomes a Kähler manifold too. It is sometimes possible to calculate its curvature tensor and show that it has negative curvature. Other people have taken this further and shown that direct images $\mathcal R^q \pi_* \mathcal F$ of sheaves on $\mathcal X$ tend to have negatively curved metrics too.
So what does this give us? What do we know now about manifolds like $X$ that we didn't know before? How does knowing that its moduli space is a complex manifold let us prove things about the original manifold?
Consider for example K3 surfaces. There's a proof of the fact that those are all diffeomorphic that goes like this: There exists a (coarse) moduli space of K3 surfaces. We can show that Kummer surfaces are dense in that space, and Kummer surfaces are all diffeomorphic by inspection, so we win.
This feels very nice, but the only information about the moduli space we used here was topological. Its complex or Kähler structures didn't play a role at all. What is a problem where they are important?