I am currently learning the basic theory of modular forms. In the theory, we identify the fundamental region
$$ D = \{ z : -1/2 \leq \Re(z) \leq 1/2, |z| \geq 1 \} $$
We can identify $\mathbf{H}/\Gamma$ as a quotient of $D$ identifying the boundary of $D$ by reflection in the $y$ axis. The space obtained is a manifold , except at the singularity points $\omega = e^{i \pi/3}$, $\omega^2 = e^{2 i \pi/3}$, and $i$, which are exactly the points where the stabilizer of $\Gamma$ is nontrivial. Thus a nontrivial stabilizer identifies the points where the manifold is nontrivial.
We see the same results in other situations. For instance, the most basic one I can think of is the action of rotation on $\mathbf{D}$. The stabilizer at points on the disc are trivial, except at the origin. When we take the quotient space under this action, we obtain a ray extending from the origin -- the space is a one-manifold, except at the origin where the stabilizer is nontrivial.
Is there a general theory as to why this is true, perhaps in the theory of actions from Lie groups?
I think your question is based on a misunderstanding. In the standard approach to the theory of modular forms via Riemann surfaces, the quotient $\mathbf{H} / \Gamma$ is a manifold, and the images of these funny points ('elliptic points') are not singular points of the quotient. They are ramification points of the map from $\mathbf{H}$ to the quotient, which is a different thing entirely: it's a property of the map, not of the target space.
(There are alternative perspectives on this, using the language of algebraic stacks; but that's not relevant at this level.)