I want to prove the existence of a geodesic $\gamma$ with end points $\xi$ and $\xi'$ at infinity in a proper CAT(-1) space.
A hint in Elements of Asymptotic Geometry is to first notice that each geodesic segment $\xi(t) \xi'(t)$, with $t \geq 0$, intersects a fixed ball $B_R(o)$ with $R \geq 2(\xi|\xi')_o$. Because $\gamma$ is proper, the boundary at infinity coincides with the geodesic boundary. Using this, I am supposed to find $\gamma$ as a sublimit of $\xi(t) \xi'(t)$ as $t \to \infty$. Maybe this is obvious, but I am not sure why I should consider the ball of radius $R$, and the significance of the inequality $R \geq 2(\xi|\xi')_o$ for a general hyperbolic space.
Any help to start would be appreciated.
The significance of the inequality $R \ge 2(\xi \mid \xi')_o$ is, as you have written, to use it in order to verify that each geodesic segment $\xi(t) \xi'(t)$ intersects the ball $B_R(0)$.
The significance of knowing that the ball $B_R(0)$ intersects $\xi(t)\xi'(t)$ for all $t \ge 0$ is that you can use that, together with properness of the space, to conclude that $B_R(0)$ is (pre)compact, using which you can then verify that the hypotheses of the Ascoli-Arzela theorem are satisfied.
So now you draw some conclusions from that theorem. The first conclusion is that there exists a sequence $t_n$ with $t_n \to +\infty$ such that the intersection of $\xi(t_n) \xi'(t_n)$ with $B_R(0)$ converges to a geodesic segment $\ell_R \subset B_R(0)$ with endpoints on the boundary of $B_R(0)$.
Next, you prove that there is a subsequence of $t_n$ such that the intersection of the subsequence of subsegments with $B_{2R}(0)$ converges to a geodesic segment $\ell_{2R} \subset B_{2R}(0)$ with endpoints on the boundary of $B_{2R}(0)$; and furthermore $\ell_R \subset \ell_{2R}$.
You continue in this manner by induction, extracting subsequence after subsequence to get convergence to geodesic segments $\ell_{kR}$ in larger and larger balls $B_{kR}(0)$, with endpoints on the boundary of each ball, such that $\ell_R \subset \ell_{2R} \subset \cdots \subset \ell_{kR} \subset \cdots$.
Taking the union as $k \to \infty$ you obtain an entire bi-infinite geodesic line $\ell$. Your final work is to verify that this geodesic has endpoints $\xi,\xi'$.