I can not say why one studies orbifolds (or e.g. why one studies math at all). However, I can try the approach which might convince your funding agency: There are tons of interesting examples of how orbifolds arise in "applications" (=mathematics):

1. The quotient spaces appearing in symplectic reduction are not always manifolds. If they fail to be manifolds, then one can (also not always but often enough) give them the structure of an orbifold (and then hope to do differential geometry on them), see Ana Cannas da Silvas notes for some interesting examples: https://people.math.ethz.ch/~acannas/Papers/lsg.pdf

2. If one is interested in shape analysis (see https://arxiv.org/pdf/1305.1150.pdf) one wants to study Riemannian geometry on quotients of the form Imm $([0,1] , \mathbb{R}^n)$ / Diff$_+$ ([0,1]) (Immersions of the interval into $\mathbb{R}^n$ mod the orientation preserving diffeomorphisms of the unit interval). Unfortunately this is not a manifold as there are immersions which may have a finite stabiliser subgroup under the reparametrisation action of Diff$_+ ([0,1])$ (this is a result of P.W. Michor and collaborators, see V. Cervera, F. Mascaro, and P. W. Michor. The action of the diffeomorphism group on the space of immersions.Differential Geom. Appl., 1(4):391–401, 1991.) So in essence, these spaces are "infinite-dimensional orbifolds" (in shape analysis this is immediatly disregarded as one then concentrates on the open subset of elements with trivial stabiliser.

3. Orbifolds appear naturally in questions connected to foliation theory (see e.g. Moerdijk/Mrcun: Introduction to Foliations and Lie Groupoids)

4. Thurston studied them in his work on geometrisation of $3$-dimensional manifolds (see http://library.msri.org/books/gt3m/)

Though this is by no means an exhaustive list, I hope that there is some example you find interesting enough to justify interest in orbifolds.

I can not say why one studies orbifolds (or e.g. why one studies math at all). However, I can try the approach which might convince your funding agency: There are tons of interesting examples of how orbifolds arise in "applications" (= mathematics):

1. The quotient spaces appearing in symplectic reduction are not always manifolds. If they fail to be manifolds, then one can (also not always but often enough) give them the structure of an orbifold (and then hope to do differential geometry on them), see Ana Cannas da Silvas's notes for some interesting examples: da Silvas - Lectures on symplectic geometry.

2. If one is interested in shape analysis (see Bauer, Bruveris, and Michor - Overview of the geometries of shape spaces and diffeomorphism groups), one wants to study Riemannian geometry on quotients of the form $\operatorname{Imm}([0,1] , \mathbb{R}^n) / \operatorname{Diff}_+([0,1])$ (Immersions of the interval into $\mathbb{R}^n$ mod the orientation preserving diffeomorphisms of the unit interval). Unfortunately this is not a manifold as there are immersions which may have a finite stabiliser subgroup under the reparametrisation action of $\operatorname{Diff}_+ ([0,1])$ (this is a result of P.W. Michor and collaborators, see V. Cervera, F. Mascaro, and P. W. Michor. The action of the diffeomorphism group on the space of immersions. Differential Geom. Appl., 1(4):391–401, 1991 (MSN)). So in essence, these spaces are "infinite-dimensional orbifolds" (in shape analysis this is immediately disregarded as one then concentrates on the open subset of elements with trivial stabiliser).

3. Orbifolds appear naturally in questions connected to foliation theory (see, e.g., Moerdijk/Mrčun: Introduction to Foliations and Lie Groupoids (MSN)).

4. Thurston studied them in his work on geometrisation of $3$-dimensional manifolds (see Thurston - Geometry and topology of 3-manifolds)

Though this is by no means an exhaustive list, I hope that there is some example you find interesting enough to justify interest in orbifolds.