Book: An introduction to Teichmuller spaces by Imayoshi & Taniguchi.
Let $R$ be a Riemann surface whose universal cover is $\Bbb D^2$ (i.e. Hyperbolic surface). Consider cutting $R$ by a family of mutually disjoint simple closed geodesics on $R$. Let $P$ be a relatively compact connected component of the resulting union of subsurfaces that doesn't contain any more simple closed geometric of $R$. Then $P$ is a triply connected surface (pants).
Fix a pants $P$ of $R$ arbitrarily. Let $\Gamma$ be a Fuchsian group corresponding to $R$ so that $\pi:\Bbb D^2\to R = \Bbb D^2/\Gamma$ is a covering projection. Let $\tilde{P}$ be a connected component of $\pi^{-1}(P)$. Denote by $\Gamma_{\tilde{P}}$ the subgroup of $\Gamma$ consisting of all elements $\gamma$ of $\Gamma$ such that $\gamma(\tilde{P}) = \tilde{P}$. $\color{red}{\text{Then $\Gamma_{\tilde{P}}$ is a free group generated by two hyperbolic transformations}}$ and $P = \tilde{P}/\Gamma_{\tilde{P}}$.
The question is the red part. You can see the example figure below given in the text I pasted. I think the action given in the figure is moving/gluing two adjacent circular arcs so that after the quotient, boundary circles $L_i$s can be formed. But anyway I can't understand why $\Gamma_{\tilde{P}}$ is generated by two hyperbolic transformations.
Actually I have one more question. In the figure, $\hat{P} :=\Bbb D^2/\Gamma_{\tilde{P}}$, why the white part in $\Bbb D^2$ disappeared? I mean only gray and dark gray parts are on the image.
The point is that the fundamental group of $P$ (a pair of pants) is free with two generators ( $L_1,L_2$ in your picture). This generators are hyperbolic elements because we started with a compact surface (no parabolic, no elliptic elements in the fundamental group).
However, one need first to check that the group $\Gamma _p$ is a subgroup of $\pi _1(S)$.
If you choose your pair of pants with a geodesic boundary (i.e. $L_1,L_2,L_3$ are geodesics) then $\tilde P$ is a convex set (bounded by lines), so it is simply connected, and the projection onto $\tilde P$ is a $\Gamma _P$ equivariant projection $D_2\to \tilde {P}$.
The grey part is just a fundamental domain of the action of $\Gamma _p$ on the complementary of this convex set $\tilde P$.