Question is as in the title.

Why study orbifolds?

I study orbifolds as locally compact Hausdorff spaces $X$ having an orbifold structure, i.e., there exists an orbifold groupoid (proper foliatio. Groupoid) $\mathcal{G}$ and a homeomorphism $|\mathcal{G}|\rightarrow X$, where $|\mathcal{G}|$ is orbit space of the groupoid $\mathcal{G}$.

Let $G$ be a Lie group acting smoothly on a manifold $M$. If one further assumes the action is proper and free then, the quotient space $M/G$ has a manifold structure.

One reason why I find orbifolds interesting is, loosely, if I ignore the condition that the group $G$ (compact Lie group) freely and impose that it only acts almost freely, then there is still some interesting structure on the quotient space $M/G$, i.e., of orbifold.

I don’t know any other places where one would see Orbifolds and how it would be interesting.

Correct me if I have a wrong understand and share your opinion of how to see an orbifold.

Edit : Wikipedia page does not say anything about groupoid way of looking at orbifolds. I did not mean to ask for references for orbifolds or groupoids. It is about how do you explain others what orbifolds are and how they occur naturally and what tools do we use to study thei geometry.