I'm reading this paper. On page 27 near the middle the author writes
Since $\pi_1(B)$ for the pair of pants $B$ is the free group on two generators $\mathbb{Z}*\mathbb{Z}$, a principal $G$-bundle $\zeta$ is the same as a conjugacy class of a pair $(g,h)\in G\times G$.
I know why $\pi_1(B)$ is $\mathbb{Z}*\mathbb{Z}$ but why does that mean that a principal $G$-bundle $\zeta$ is the same as a conjugacy class of a pair $(g,h)\in G\times G$?
The classification of bundles on a space depends only on the homotopy type of the space, you can replace the surface by a wedge of two circles. Now a bundle over such a wedge is constructed in an obvious way from two bundles over the circle. That leaves you with the problem of classifying bundles over a circle. That can be done using the clutching construction, for example.