I am having some difficulties understanding an argument in a proof. Here is the portion from a complex dynamics paper by Lyubich-Peters:

The following argument is a more geometric presentation of the argument given in the main body of the paper. To fix the idea, let us assume that the image of f is the standard cusp {$w^2$ = $z^3$}. Let us include into holomorphic foliation $\mathbb{F}$ with leaves $L_{λ}$ = {$w^2$ = $λz^3$}, $λ ∈ \hat{\mathbb{C}}$. Let us puncture out 0, and consider the space $\mathbb{O}$ of leaves in the punctured neighborhood of the origin. This space has a natural Riemann orbifold structure (supported on the sphere) whose local charts are obtained by taking local transversals to F and slicing the leaves to it. There are two orbifold points on O: the leaf w = 0 is an orbifold point of order 3 and the leaf z = 0 is an orbifold point of order 2. So, the Euler characteristic of O is equal to 1/2+1/3 < 1.

I should mention that I have an introductory understanding of Orbifolds, Riemannian Orbifolds, Foliations. I am trying to understand how $\mathbb{O}$ is acquiring the structure, what supported on a sphere refers to in this case. If possible, an intuitive description of this process is what I would like to have as an answer.

P.S. By cusp they mean $\{(z,w): z^{2} - w^{3}= 0 \} \subset \mathbb{C}^{2}$. Thanks!