Here is one example. You may have heard about the 3-dimensional Poincare Conjecture (if $M$ is a compact, without boundary, simply-connected 3-dimensional manifold, then $M$ is homeomorphic to $S^3$). Here is how it was proven by Gregory Perelman, broadly speaking:
$M$ is known to admit a smooth structure. Equip $M$, therefore with a randomly chosen Riemannian metric $g$. Since we know nothing about the properties of $g$, we cannot deduce any conclusions about $M$ from the existence of $g$.
Deform $g$ via the Ricc flow (with surgeries) to an Einstein metric $g_0$ on $M$. (OK, not on $M$ itself but on pieces of a connected sum decomposition of $M$, but let's ignore this.)
In dimension 3 each Einstein metric has constant curvature. It is easy to see that this curvature has to be positive (since $M$ is compact and simply connected), hence, after rescaling, we can assume it to be equal to $1$. Hence, by the Killing-Hopf theorem (I always thought of it is Elie Cartan's theorem...) $(M, g_0)$ is isometric to the unit 3-dimensional sphere with its standard metric. In particular, $M$ itself is diffeomorphic to $S^3$.
Now, you see how this theorem is useful and what advantage one has having a metric of constant curvature on a manifold.